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<title>Intersection (geometry)</title>

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Intersection (geometry)

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</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Intersection_(mathematics)" class="mw-redirect" title="Intersection (mathematics)">Intersection (mathematics)</a>.</div>

In <a href="Geometry" title="Geometry">geometry</a>, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is the <a href="Line%E2%80%93line_intersection" title="Line–line intersection">line–line intersection</a> between two distinct <a href="Line_(geometry)" title="Line (geometry)">lines</a>, which either is one <a href="Point_(geometry)" title="Point (geometry)">point</a> (sometimes called a <a href="Vertex_(geometry)" title="Vertex (geometry)">vertex</a>) or does not exist (if the lines are <a href="Parallel_lines" class="mw-redirect" title="Parallel lines">parallel</a>). Other types of geometric intersection include:

<ul><li><a href="Line%E2%80%93plane_intersection" title="Line–plane intersection">Line–plane intersection</a></li>

<li><a href="Line%E2%80%93sphere_intersection" title="Line–sphere intersection">Line–sphere intersection</a></li>

<li><a href="Intersection_of_a_polyhedron_with_a_line" title="Intersection of a polyhedron with a line">Intersection of a polyhedron with a line</a></li>

<li><a href="Line_segment_intersection" class="mw-redirect" title="Line segment intersection">Line segment intersection</a></li>

<li><a href="Intersection_curve" title="Intersection curve">Intersection curve</a></li></ul>

Determination of the intersection of <a href="Flat_(geometry)" title="Flat (geometry)">flats</a> – linear geometric objects embedded in a higher-<a href="Dimension" title="Dimension">dimensional</a> space – is a simple task of <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, namely the solution of a <a href="System_of_linear_equations" title="System of linear equations">system of linear equations</a>. In general the determination of an intersection leads to <a href="Non-linear_equation" class="mw-redirect" title="Non-linear equation">non-linear equations</a>, which can be <a href="Numerical_solution" class="mw-redirect" title="Numerical solution">solved numerically</a>, for example using <a href="Newton_iteration" class="mw-redirect" title="Newton iteration">Newton iteration</a>. Intersection problems between a line and a <a href="Conic_section" title="Conic section">conic section</a> (circle, ellipse, parabola, etc.) or a <a href="Quadric" title="Quadric">quadric</a> (sphere, cylinder, hyperboloid, etc.) lead to <a href="Quadratic_equation" title="Quadratic equation">quadratic equations</a> that can be easily solved. Intersections between quadrics lead to <a href="Quartic_equation" title="Quartic equation">quartic equations</a> that can be solved <a href="Algebraic_equation" title="Algebraic equation">algebraically</a>.

<div class="mw-heading mw-heading2"><h2 id="On_a_plane">On a plane</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane (geometry)</a> and <a href="Two-dimensional_space" title="Two-dimensional space">Two-dimensional space</a></div>

<div class="mw-heading mw-heading3"><h3 id="Two_lines">Two lines</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Line%E2%80%93line_intersection" title="Line–line intersection">Line–line intersection</a></div>

For the determination of the intersection point of two non-parallel lines

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<annotation encoding="application/x-tex">{\displaystyle a_{1}x+b_{1}y=c_{1},\ a_{2}x+b_{2}y=c_{2}}</annotation>

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</math><img src="./744001a90bb1b9d716a6dbb24fcb8cac444e0a96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.256ex; height:2.509ex;" alt="{\displaystyle a_{1}x+b_{1}y=c_{1},\ a_{2}x+b_{2}y=c_{2}}" loading="lazy">

one gets, from <a href="Cramer's_rule" title="Cramer's rule">Cramer's rule</a> or by substituting out a variable, the coordinates of the intersection point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{s},y_{s})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{s},y_{s})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle x_{s}={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}},\quad y_{s}={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}.\ }</annotation>

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</math><img src="./3db2304cc7f523a02ff2f1ca1629505004538ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.359ex; height:5.843ex;" alt="{\displaystyle x_{s}={\frac {c_{1}b_{2}-c_{2}b_{1}}{a_{1}b_{2}-a_{2}b_{1}}},\quad y_{s}={\frac {a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}}}.\ }" loading="lazy"></dd></dl>

(If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}b_{2}-a_{2}b_{1}=0}">

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<annotation encoding="application/x-tex">{\displaystyle a_{1}b_{2}-a_{2}b_{1}=0}</annotation>

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</math><img src="./98a97d386948628ba33d05691594288c17c6a95a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.773ex; height:2.509ex;" alt="{\displaystyle a_{1}b_{2}-a_{2}b_{1}=0}" loading="lazy"> the lines are parallel and these formulas cannot be used because they involve dividing by 0.)

<div class="mw-heading mw-heading3"><h3 id="Two_line_segments">Two line segments</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Multiple_line_segment_intersection" title="Multiple line segment intersection">Multiple line segment intersection</a> and <a href="Line%E2%80%93line_intersection#Given_two_points_on_each_line_segment" title="Line–line intersection">Line–line intersection § Given two points on each line segment</a></div>

For two non-parallel <a href="Line_segment" title="Line segment">line segments</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1}),(x_{2},y_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1}),(x_{2},y_{2})}</annotation>

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</math><img src="./fac3e5f2220d5c90f37b29cb9ee8fda5517872ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.875ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1}),(x_{2},y_{2})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{3},y_{3}),(x_{4},y_{4})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{3},y_{3}),(x_{4},y_{4})}</annotation>

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</math><img src="./1ea07723eeaea0dacdfadbfdcbc64e7659c9fce9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.875ex; height:2.843ex;" alt="{\displaystyle (x_{3},y_{3}),(x_{4},y_{4})}" loading="lazy"> there is not necessarily an intersection point (see diagram), because the intersection point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation>

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</math><img src="./29c296094af9a1c665425debeac5eaab99a37a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" loading="lazy"> of the corresponding lines need not to be contained in the line segments. In order to check the situation one uses parametric representations of the lines:

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<annotation encoding="application/x-tex">{\displaystyle (x(s),y(s))=(x_{1}+s(x_{2}-x_{1}),y_{1}+s(y_{2}-y_{1})),}</annotation>

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</math><img src="./5408b018255895a0386c6cb4e25097dbdd44ea08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.61ex; height:2.843ex;" alt="{\displaystyle (x(s),y(s))=(x_{1}+s(x_{2}-x_{1}),y_{1}+s(y_{2}-y_{1})),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle (x(t),y(t))=(x_{3}+t(x_{4}-x_{3}),y_{3}+t(y_{4}-y_{3})).}</annotation>

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</math><img src="./67e1a16edd80a1c2171b3f5333daa4fd72987163.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.606ex; height:2.843ex;" alt="{\displaystyle (x(t),y(t))=(x_{3}+t(x_{4}-x_{3}),y_{3}+t(y_{4}-y_{3})).}" loading="lazy"></dd></dl>

The line segments intersect only in a common point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation>

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</math><img src="./29c296094af9a1c665425debeac5eaab99a37a04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}" loading="lazy"> of the corresponding lines if the corresponding parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0},t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{0},t_{0}}</annotation>

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</math><img src="./59687b292bceb721abe01f5f3169a8b49af39f65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.072ex; height:2.343ex;" alt="{\displaystyle s_{0},t_{0}}" loading="lazy"> fulfill the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq s_{0},t_{0}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq s_{0},t_{0}\leq 1}</annotation>

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The parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0},t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{0},t_{0}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle s(x_{2}-x_{1})-t(x_{4}-x_{3})=x_{3}-x_{1},}</annotation>

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</math><img src="./eac35e7c95898b6ee8df54738353c27be2b01c7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.959ex; height:2.843ex;" alt="{\displaystyle s(x_{2}-x_{1})-t(x_{4}-x_{3})=x_{3}-x_{1},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle s(y_{2}-y_{1})-t(y_{4}-y_{3})=y_{3}-y_{1}\ .}</annotation>

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</math><img src="./95ba9416d48755bd20b96b16276ff83c4b1d98f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.397ex; height:2.843ex;" alt="{\displaystyle s(y_{2}-y_{1})-t(y_{4}-y_{3})=y_{3}-y_{1}\ .}" loading="lazy"></dd></dl>

It can be solved for s and t using Cramer's rule (see <a href="#Two_lines">above</a>). If the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq s_{0},t_{0}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq s_{0},t_{0}\leq 1}</annotation>

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</math><img src="./1e15203966faab7b196460cdad6cba69f3e62894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.594ex; height:2.509ex;" alt="{\displaystyle 0\leq s_{0},t_{0}\leq 1}" loading="lazy"> is fulfilled one inserts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}}">

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</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> into the corresponding parametric representation and gets the intersection point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}">

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Example: For the line segments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,1),(3,2)}">

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</math><img src="./e026db88de730051a5a7bfc567ce1b6dd8dd5a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.37ex; height:2.843ex;" alt="{\displaystyle (1,1),(3,2)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,4),(2,-1)}">

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<annotation encoding="application/x-tex">{\displaystyle (1,4),(2,-1)}</annotation>

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</math><img src="./361203a238986059704df98ec55b034fccd32152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.178ex; height:2.843ex;" alt="{\displaystyle (1,4),(2,-1)}" loading="lazy"> one gets the linear system

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<annotation encoding="application/x-tex">{\displaystyle s+5t=3}</annotation>

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</math><img src="./8a8976132d3a5c22988bb8d0f02d78a4c9106efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.194ex; height:2.343ex;" alt="{\displaystyle s+5t=3}" loading="lazy"></dd></dl>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}={\tfrac {3}{11}},t_{0}={\tfrac {6}{11}}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{0}={\tfrac {3}{11}},t_{0}={\tfrac {6}{11}}}</annotation>

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</math><img src="./c81840748c91cfe8c7e77fcd1662b18a66572b72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.23ex; height:3.509ex;" alt="{\displaystyle s_{0}={\tfrac {3}{11}},t_{0}={\tfrac {6}{11}}}" loading="lazy">. That means: the lines intersect at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\tfrac {17}{11}},{\tfrac {14}{11}})}">

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<annotation encoding="application/x-tex">{\displaystyle ({\tfrac {17}{11}},{\tfrac {14}{11}})}</annotation>

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Remark: Considering lines, instead of segments, determined by pairs of points, each condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq s_{0},t_{0}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq s_{0},t_{0}\leq 1}</annotation>

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</math><img src="./1e15203966faab7b196460cdad6cba69f3e62894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.594ex; height:2.509ex;" alt="{\displaystyle 0\leq s_{0},t_{0}\leq 1}" loading="lazy"> can be dropped and the method yields the intersection point of the lines (see <a href="#Two_lines">above</a>).

<div class="mw-heading mw-heading3"><h3 id="A_line_and_a_circle">A line and a circle</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Line%E2%80%93sphere_intersection" title="Line–sphere intersection">Line–sphere intersection</a></div>

For the intersection of

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</math><img src="./2a39c048e6c827a005e417be0e1bd5ac314e6fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle ax+by=c}" loading="lazy"> and <a href="Circle" title="Circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=r^{2}}">

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</math><img src="./37dd4f282df84a83620f71dc52345122e0e3a514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.64ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=r^{2}}" loading="lazy"></li></ul>

one solves the line equation for x or y and <a href="Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">substitutes</a> it into the equation of the circle and gets for the solution (using the formula of a quadratic equation) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1}),(x_{2},y_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1/2}={\frac {ac\pm b{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}</annotation>

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</math><img src="./7ac38978da106a834c21917d19e1762245b8355d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.578ex; height:6.676ex;" alt="{\displaystyle x_{1/2}={\frac {ac\pm b{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle y_{1/2}={\frac {bc\mp a{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}</annotation>

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</math><img src="./6c33949ceb405ea6321b0e8b853cd4ba93c6812e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.388ex; height:6.676ex;" alt="{\displaystyle y_{1/2}={\frac {bc\mp a{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}" loading="lazy"></dd></dl>

if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}>0\ .}">

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<annotation encoding="application/x-tex">{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}>0\ .}</annotation>

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</math><img src="./851f2200994998926fdf1c2589e8903041c021e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.478ex; height:3.176ex;" alt="{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}>0\ .}" loading="lazy"> If this condition holds with strict inequality, there are two intersection points; in this case the line is called a <a href="Secant_line" title="Secant line">secant line</a> of the circle, and the line segment connecting the intersection points is called a <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> of the circle.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}=0}</annotation>

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</math><img src="./de31f72e06deeaf974111998e1a1984f4c773454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.251ex; height:3.176ex;" alt="{\displaystyle r^{2}(a^{2}+b^{2})-c^{2}=0}" loading="lazy"> holds, there exists only one intersection point and the line is tangent to the circle. If the weak inequality does not hold, the line does not intersect the circle.

If the circle's midpoint is not the origin, see.<a href="#cite_note-1">[1]</a> The intersection of a line and a parabola or hyperbola may be treated analogously.

<div class="mw-heading mw-heading3"><h3 id="Two_circles">Two circles</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Lens_(geometry)" title="Lens (geometry)">Lens (geometry)</a></div>

The determination of the intersection points of two circles

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<annotation encoding="application/x-tex">{\displaystyle (x-x_{1})^{2}+(y-y_{1})^{2}=r_{1}^{2},\ \quad (x-x_{2})^{2}+(y-y_{2})^{2}=r_{2}^{2}}</annotation>

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can be reduced to the previous case of intersecting a line and a circle. By subtraction of the two given equations one gets the line equation:

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<annotation encoding="application/x-tex">{\displaystyle 2(x_{2}-x_{1})x+2(y_{2}-y_{1})y=r_{1}^{2}-x_{1}^{2}-y_{1}^{2}-r_{2}^{2}+x_{2}^{2}+y_{2}^{2}.}</annotation>

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</math><img src="./f0737d826762742ac268170d62db2582b00412ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.455ex; height:3.176ex;" alt="{\displaystyle 2(x_{2}-x_{1})x+2(y_{2}-y_{1})y=r_{1}^{2}-x_{1}^{2}-y_{1}^{2}-r_{2}^{2}+x_{2}^{2}+y_{2}^{2}.}" loading="lazy"></dd></dl>

This special line is the <a href="Radical_line" class="mw-redirect" title="Radical line">radical line</a> of the two circles.

Special case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;x_{1}=y_{1}=y_{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \;x_{1}=y_{1}=y_{2}=0}</annotation>

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In this case the origin is the center of the first circle and the second center lies on the x-axis (s. diagram). The equation of the radical line simplifies to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;2x_{2}x=r_{1}^{2}-r_{2}^{2}+x_{2}^{2}\;}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}={\frac {r_{1}^{2}-r_{2}^{2}+x_{2}^{2}}{2x_{2}}},\quad y_{0}={\sqrt {r_{1}^{2}-x_{0}^{2}}}\ .}</annotation>

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</math><img src="./4ec3d9693325a012e26081cca1268463ae0b8c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:38.116ex; height:6.176ex;" alt="{\displaystyle x_{0}={\frac {r_{1}^{2}-r_{2}^{2}+x_{2}^{2}}{2x_{2}}},\quad y_{0}={\sqrt {r_{1}^{2}-x_{0}^{2}}}\ .}" loading="lazy"></dd></dl>

In case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}^{2}<x_{0}^{2}}">

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</math><img src="./00d37246df88c33338ace234f6d796f2b4e50e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.585ex; height:3.176ex;" alt="{\displaystyle r_{1}^{2}<x_{0}^{2}}" loading="lazy"> the circles have no points in common.

In case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}^{2}=x_{0}^{2}}">

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Any general case as written above can be transformed by a shift and a rotation into the special case.

The intersection of two <a href="Disk_(mathematics)" title="Disk (mathematics)">disks</a> (the interiors of the two circles) forms a shape called a <a href="Lens_(geometry)" title="Lens (geometry)">lens</a>.

<div class="mw-heading mw-heading3"><h3 id="Two_conic_sections">Two conic sections</h3></div>

The problem of intersection of an ellipse/hyperbola/parabola with another <a href="Conic_section" title="Conic section">conic section</a> leads to a <a href="System_of_polynomial_equations" title="System of polynomial equations">system of quadratic equations</a>, which can be solved in special cases easily by elimination of one coordinate. Special properties of conic sections may be used to obtain a <a href="Conic_section#Intersecting_two_conics" title="Conic section">solution</a>. In general the intersection points can be determined by solving the equation by a Newton iteration. If a) both conics are given implicitly (by an equation) a 2-dimensional Newton iteration b) one implicitly and the other parametrically given a 1-dimensional Newton iteration is necessary. See next section.

<div class="mw-heading mw-heading3"><h3 id="Two_smooth_curves">Two smooth curves</h3></div>

Two curves in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

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have an intersection point, if they have a point of the plane in common and have at this point (see diagram):

<dl><dd>a) different tangent lines (transversal intersection, after <a href="Transversality_(mathematics)" class="mw-redirect" title="Transversality (mathematics)">transversality</a>), or</dd>

<dd>b) the tangent line in common and they are crossing each other (touching intersection, after <a href="Tangency" class="mw-redirect" title="Tangency">tangency</a>).</dd></dl>

If both the curves have a point S and the tangent line there in common but do not cross each other, they are just touching at point S.

Because touching intersections appear rarely and are difficult to deal with, the following considerations omit this case. In any case below all necessary differential conditions are presupposed. The determination of intersection points always leads to one or two non-linear equations which can be solved by Newton iteration. A list of the appearing cases follows:

<ul><li>If both curves are explicitly given: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f_{1}(x),\ y=f_{2}(x)}">

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<ul><li>If both curves are parametrically given: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}:(x_{1}(t),y_{1}(t)),\ C_{2}:(x_{2}(s),y_{2}(s)).}">

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<dl><dd>Equating them yields two equations in two variables:

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<ul><li>If one curve is parametrically and the other implicitly given: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}:(x_{1}(t),y_{1}(t)),\ C_{2}:f(x,y)=0.}">

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<dl><dd>This is the simplest case besides the explicit case. One has to insert the parametric representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x_{1}(t),y_{2}(t))=0\ .}</annotation>

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<ul><li>If both curves are implicitly given: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}:f_{1}(x,y)=0,\ C_{2}:f_{2}(x,y)=0.}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}:f_{1}(x,y)=0,\ C_{2}:f_{2}(x,y)=0.}</annotation>

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<dl><dd>Here, an intersection point is a solution of the system

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<annotation encoding="application/x-tex">{\displaystyle f_{1}(x,y)=0,\ f_{2}(x,y)=0\ .}</annotation>

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Any Newton iteration needs convenient starting values, which can be derived by a visualization of both the curves. A parametrically or explicitly given curve can easily be visualized, because to any parameter t or x respectively it is easy to calculate the corresponding point. For implicitly given curves this task is not as easy. In this case one has to determine a curve point with help of starting values and an iteration. See

.<a href="#cite_note-2">[2]</a>

Examples:

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<annotation encoding="application/x-tex">{\displaystyle C_{2}:(x-1)^{2}+(y-1)^{2}-10=0}</annotation>

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</math><img src="./6636f8567d8e1a8fffccf0d321d42f526605764b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.138ex; height:3.176ex;" alt="{\displaystyle C_{2}:(x-1)^{2}+(y-1)^{2}-10=0}" loading="lazy"> (see diagram).

<dl><dd>The Newton iteration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n+1}:=t_{n}-{\frac {f(t_{n})}{f'(t_{n})}}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{n+1}:=t_{n}-{\frac {f(t_{n})}{f'(t_{n})}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=(t-1)^{2}+(t^{3}-1)^{2}-10}</annotation>

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</math><img src="./3a63743c006b1d7916f707c67bfc3bb62262e6b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.498ex; height:3.176ex;" alt="{\displaystyle f(t)=(t-1)^{2}+(t^{3}-1)^{2}-10}" loading="lazy"> has to be done. As start values one can choose −1 and 1.5.</dd></dl></dd>

<dd>The intersection points are: (−1.1073, −1.3578), (1.6011, 4.1046)</dd></dl></dd>

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<annotation encoding="application/x-tex">{\displaystyle C_{1}:f_{1}(x,y)=x^{4}+y^{4}-1=0,}</annotation>

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</math><img src="./3fae90ff04922b933d426e4fbfa57a49a82e219c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.623ex; height:3.176ex;" alt="{\displaystyle C_{1}:f_{1}(x,y)=x^{4}+y^{4}-1=0,}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle C_{2}:f_{2}(x,y)=(x-0.5)^{2}+(y-0.5)^{2}-1=0}</annotation>

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<dd>The Newton iteration

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<annotation encoding="application/x-tex">{\displaystyle {x_{n+1} \choose y_{n+1}}={x_{n}+\delta _{x} \choose y_{n}+\delta _{y}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\delta _{x} \choose \delta _{y}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}{\frac {\partial f_{1}}{\partial x}}&{\frac {\partial f_{1}}{\partial y}}\\{\frac {\partial f_{2}}{\partial x}}&{\frac {\partial f_{2}}{\partial y}}\end{pmatrix}}{\delta _{x} \choose \delta _{y}}={-f_{1} \choose -f_{2}}}</annotation>

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</math><img src="./0cb72531c2eadf8fb75773f96632220d93b6df3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.884ex; margin-bottom: -0.287ex; width:30.099ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}{\frac {\partial f_{1}}{\partial x}}&{\frac {\partial f_{1}}{\partial y}}\\{\frac {\partial f_{2}}{\partial x}}&{\frac {\partial f_{2}}{\partial y}}\end{pmatrix}}{\delta _{x} \choose \delta _{y}}={-f_{1} \choose -f_{2}}}" loading="lazy"> at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n},y_{n})}">

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<dd>The linear system can be solved by Cramer's rule.</dd>

<dd>The intersection points are (−0.3686, 0.9953) and (0.9953, −0.3686).</dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Two_polygons">Two polygons</h3></div>

If one wants to determine the intersection points of two <a href="Polygon" title="Polygon">polygons</a>, one can check the intersection of any pair of line segments of the polygons (see <a href="#Two_line_segments">above</a>). For polygons with many segments this method is rather time-consuming. In practice one accelerates the intersection algorithm by using window tests. In this case one divides the polygons into small sub-polygons and determines the smallest window (rectangle with sides parallel to the coordinate axes) for any sub-polygon. Before starting the time-consuming determination of the intersection point of two line segments any pair of windows is tested for common points. See.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="In_space_(three_dimensions)">In space (three dimensions)</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a></div>

In 3-dimensional space there are intersection points (common points) between curves and surfaces. In the following sections we consider <a href="Transversality_(mathematics)" class="mw-redirect" title="Transversality (mathematics)">transversal</a> intersection only.

<div class="mw-heading mw-heading3"><h3 id="A_line_and_a_plane">A line and a plane</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Line%E2%80%93plane_intersection" title="Line–plane intersection">Line–plane intersection</a></div>

The intersection of a line and a plane in <a href="General_position" title="General position">general position</a> in three dimensions is a point.

Commonly a line in space is represented parametrically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x(t),y(t),z(t))}">

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for parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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If the linear equation has no solution, the line either lies on the plane or is parallel to it.

<div class="mw-heading mw-heading3"><h3 id="Three_planes">Three planes</h3></div>

If a line is defined by two intersecting planes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}:\ {\vec {n}}_{i}\cdot {\vec {x}}=d_{i},\ i=1,2}">

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Three planes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}:\ {\vec {n}}_{i}\cdot {\vec {x}}=d_{i},\ i=1,2,3}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {p}}_{0}={\frac {d_{1}({\vec {n}}_{2}\times {\vec {n}}_{3})+d_{2}({\vec {n}}_{3}\times {\vec {n}}_{1})+d_{3}({\vec {n}}_{1}\times {\vec {n}}_{2})}{{\vec {n}}_{1}\cdot ({\vec {n}}_{2}\times {\vec {n}}_{3})}}\ .}</annotation>

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</math><img src="./adf4682b13a945c34c0dfb98cd3977704fbb4897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:48.744ex; height:6.509ex;" alt="{\displaystyle {\vec {p}}_{0}={\frac {d_{1}({\vec {n}}_{2}\times {\vec {n}}_{3})+d_{2}({\vec {n}}_{3}\times {\vec {n}}_{1})+d_{3}({\vec {n}}_{1}\times {\vec {n}}_{2})}{{\vec {n}}_{1}\cdot ({\vec {n}}_{2}\times {\vec {n}}_{3})}}\ .}" loading="lazy"></dd></dl>

For the proof one should establish <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}_{i}\cdot {\vec {p}}_{0}=d_{i},\ i=1,2,3,}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}_{i}\cdot {\vec {p}}_{0}=d_{i},\ i=1,2,3,}</annotation>

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</math><img src="./009f99b8ff31fa66fc3d1a62f68a58b69d63280e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.078ex; height:2.843ex;" alt="{\displaystyle {\vec {n}}_{i}\cdot {\vec {p}}_{0}=d_{i},\ i=1,2,3,}" loading="lazy"> using the rules of a <a href="Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a>. If the scalar triple product equals to 0, then planes either do not have the triple intersection or it is a line (or a plane, if all three planes are the same).

<div class="mw-heading mw-heading3"><h3 id="A_curve_and_a_surface">A curve and a surface</h3></div>

Analogously to the plane case the following cases lead to non-linear systems, which can be solved using a 1- or 3-dimensional Newton iteration.<a href="#cite_note-4">[4]</a>

<ul><li>parametric curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:(x(t),y(t),z(t))}">

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<annotation encoding="application/x-tex">{\displaystyle C:(x(t),y(t),z(t))}</annotation>

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</math><img src="./cbdb62c7b61a8f9866ed9b3454f0040496a1eae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.101ex; height:2.843ex;" alt="{\displaystyle C:(x(t),y(t),z(t))}" loading="lazy"> and</li></ul>

<dl><dd>parametric surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:(x(u,v),y(u,v),z(u,v))\ ,}">

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<annotation encoding="application/x-tex">{\displaystyle S:(x(u,v),y(u,v),z(u,v))\ ,}</annotation>

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</math><img src="./8da2d4bfc41806b42621e53e98dc6d8d241ddaaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.016ex; height:2.843ex;" alt="{\displaystyle S:(x(u,v),y(u,v),z(u,v))\ ,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle C:(x(t),y(t),z(t))}</annotation>

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<dl><dd>implicit surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:f(x,y,z)=0\ .}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S:f(x,y,z)=0\ .}</annotation>

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</math><img src="./92b0a8d4765551fa56c021d0fb541d71e7054983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.654ex; height:2.843ex;" alt="{\displaystyle S:f(x,y,z)=0\ .}" loading="lazy"></dd></dl>

Example:

<dl><dd>parametric curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:(t,t^{2},t^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle C:(t,t^{2},t^{3})}</annotation>

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</math><img src="./e0736ba0002f44e982aa8479186ef80b1af206d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.208ex; height:3.176ex;" alt="{\displaystyle C:(t,t^{2},t^{3})}" loading="lazy"> and</dd>

<dd>implicit surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:x^{4}+y^{4}+z^{4}-1=0}">

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<annotation encoding="application/x-tex">{\displaystyle S:x^{4}+y^{4}+z^{4}-1=0}</annotation>

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</math><img src="./383eef0d4a67d1e7ceba0e58bf14eaed81f536fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.124ex; height:3.009ex;" alt="{\displaystyle S:x^{4}+y^{4}+z^{4}-1=0}" loading="lazy"> (s. picture).</dd>

<dd>The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332).</dd></dl>

A <a href="Line%E2%80%93sphere_intersection" title="Line–sphere intersection">line–sphere intersection</a> is a simple special case.

Like the case of a line and a plane, the intersection of a curve and a surface in <a href="General_position" title="General position">general position</a> consists of discrete points, but a curve may be partly or totally contained in a surface.

<div class="mw-heading mw-heading3"><h3 id="A_line_and_a_polyhedron">A line and a polyhedron</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Intersection_of_a_polyhedron_with_a_line" title="Intersection of a polyhedron with a line">Intersection of a polyhedron with a line</a></div>

<div class="mw-heading mw-heading3"><h3 id="Two_surfaces">Two surfaces</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Intersection_curve" title="Intersection curve">Intersection curve</a></div>

Two transversally intersecting surfaces give an <a href="Intersection_curve" title="Intersection curve">intersection curve</a>. The most simple case is the intersection line of two non-parallel planes.

<div class="mw-heading mw-heading3"><h3 id="A_sphere_and_a_plane">A sphere and a plane</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Spherical_circle" title="Spherical circle">Spherical circle</a></div>

When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:

Let S be a sphere with center O, P a plane which intersects S. Draw OE perpendicular to P and meeting P at E. Let A and B be any two different points in the intersection. Then AOE and BOE are right triangles with a common side, OE, and hypotenuses AO and BO equal. Therefore, the remaining sides AE and BE are equal. This proves that all points in the intersection are the same distance from the point E in the plane P, in other words all points in the intersection lie on a circle C with center E.<a href="#cite_note-5">[5]</a> This proves that the intersection of P and S is contained in C. Note that OE is the axis of the circle.

Now consider a point D of the circle C. Since C lies in P, so does D. On the other hand, the triangles AOE and DOE are right triangles with a common side, OE, and legs EA and ED equal. Therefore, the hypotenuses AO and DO are equal, and equal to the radius of S, so that D lies in S. This proves that C is contained in the intersection of P and S.

As a corollary, on a sphere there is exactly one circle that can be drawn through three given points.<a href="#cite_note-6">[6]</a>

The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.<a href="#cite_note-7">[7]</a>

Compare also <a href="Conic_section" title="Conic section">conic sections</a>, which can produce <a href="Oval" title="Oval">ovals</a>.

<div class="mw-heading mw-heading3"><h3 id="Two_spheres">Two spheres</h3></div>

To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">) is centered at the origin. Points on this sphere satisfy

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.}</annotation>

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</math><img src="./7b6d806e8fc81e2929f5dfbe9e3933d6865c6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.988ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2}.}" loading="lazy"></dd></dl>

Also without loss of generality, assume that the second sphere, with radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, is centered at a point on the positive x-axis, at distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> from the origin. Its points satisfy

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<annotation encoding="application/x-tex">{\displaystyle (x-a)^{2}+y^{2}+z^{2}=r^{2}.}</annotation>

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</math><img src="./cbe27af46a70a6b9c372b474c2166b0a99caf289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.152ex; height:3.176ex;" alt="{\displaystyle (x-a)^{2}+y^{2}+z^{2}=r^{2}.}" loading="lazy"></dd></dl>

The intersection of the spheres is the set of points satisfying both equations. Subtracting the equations gives

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(x-a)^{2}-x^{2}&=r^{2}-R^{2}\\a^{2}-2ax&=r^{2}-R^{2}\\x&={\frac {a^{2}+R^{2}-r^{2}}{2a}}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./55d447a56b633f7ecde4f58289afd3a74592142d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:31.707ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}(x-a)^{2}-x^{2}&=r^{2}-R^{2}\\a^{2}-2ax&=r^{2}-R^{2}\\x&={\frac {a^{2}+R^{2}-r^{2}}{2a}}.\end{aligned}}}" loading="lazy"></dd></dl>

In the singular case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=0}</annotation>

</semantics>

</math><img src="./90d476e5e765a5d77bbcff32e4584579207ec7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}" loading="lazy">, the spheres are concentric. There are two possibilities: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=r}</annotation>

</semantics>

</math><img src="./4ed97a934a158af5ecdf7587f66ec24b3234f52f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle R=r}" loading="lazy">, the spheres coincide, and the intersection is the entire sphere; if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\not =r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\not =r}</annotation>

</semantics>

</math><img src="./0b19c354c4c4ef85ac47c78d4adb6aa4e1af6d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.676ex;" alt="{\displaystyle R\not =r}" loading="lazy">, the spheres are disjoint and the intersection is empty.

When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.

The result follows from the previous proof for sphere-plane intersections.

<ul><li><a href="Line-plane_intersection" class="mw-redirect" title="Line-plane intersection">Line-plane intersection</a></li>

<li><a href="Line%E2%80%93sphere_intersection" title="Line–sphere intersection">Line–sphere intersection</a></li>

<li><a href="Line-cylinder_intersection" title="Line-cylinder intersection">Line-cylinder intersection</a></li>

<li><a href="Analytic_geometry#Intersections" title="Analytic geometry">Analytic geometry#Intersections</a></li>

<li><a href="Computational_geometry" title="Computational geometry">Computational geometry</a></li>

<li><a href="Equation_of_a_line" class="mw-redirect" title="Equation of a line">Equation of a line</a></li>

<li><a href="Intersection_(set_theory)" title="Intersection (set theory)">Intersection (set theory)</a></li>

<li><a href="Intersection_theory" title="Intersection theory">Intersection theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Erich Hartmann: <a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf">Geometry and Algorithms for COMPUTER AIDED DESIGN</a>. Lecture notes, Technische Universität Darmstadt, October 2003, p. 17

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Erich Hartmann: <a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf">Geometry and Algorithms for COMPUTER AIDED DESIGN</a>. Lecture notes, Technische Universität Darmstadt, October 2003, p. 33

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Erich Hartmann: <a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdg-skript-1998.pdf">CDKG: Computerunterstützte Darstellende und Konstruktive Geometrie</a>. Lecture notes, TU Darmstadt, 1997, p. 79 (PDF; 3,4 MB)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Erich Hartmann: <a rel="nofollow" class="external text" href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf">Geometry and Algorithms for COMPUTER AIDED DESIGN</a>. Lecture notes, Technische Universität Darmstadt, October 2003, p. 93

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Proof follows Hobbs, Prop. 304

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Hobbs, Prop. 308

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Hobbs, Prop. 310

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHobbs1921" class="citation book cs1">Hobbs, C.A. (1921). <a rel="nofollow" class="external text" href="https://archive.org/details/solidgeometry00hobbgoog">Solid Geometry</a>. G.H. Kent. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/solidgeometry00hobbgoog/page/n151">397</a> ff.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFHaines2021" class="citation web cs1">Haines, Eric (June 6, 2021). <a rel="nofollow" class="external text" href="https://www.realtimerendering.com/intersections.html">"Intersections (Ray Tracing Resources Page)"</a>. Real-Time Rendering. Retrieved December 14, 2023. <q>a grid of intersection routines for various popular objects, pointing to resources in books and on the web.</q></cite></li>

<li>Nicholas M. Patrikalakis and Takashi Maekawa, Shape Interrogation for Computer Aided Design and Manufacturing, Springer, 2002, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3540424547</bdi>, 9783540424543, pp. 408. <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=qoxBeti5ySoC&q=intersection">[1]</a></li>

<li><cite id="CITEREFSykesComstock1922" class="citation book cs1">Sykes, M.; Comstock, C.E. (1922). <a rel="nofollow" class="external text" href="https://archive.org/details/solidgeometry01comsgoog">Solid Geometry</a>. Rand McNally. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/solidgeometry01comsgoog/page/n93">81</a> ff.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-09-10" href="https://en.wikipedia.org/wiki/?title=Intersection_(geometry)&oldid=1245067921">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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