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<title>Line–plane intersection</title>

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Line–plane intersection

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In analytic <a href="Geometry" title="Geometry">geometry</a>, the intersection of a <a href="Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> and a <a href="Plane_(mathematics)" title="Plane (mathematics)">plane</a> in <a href="Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a> can be the <a href="Empty_set" title="Empty set">empty set</a>, a <a href="Point_(geometry)" title="Point (geometry)">point</a>, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in <a href="Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="Motion_planning" title="Motion planning">motion planning</a>, and <a href="Collision_detection" title="Collision detection">collision detection</a>.

<div class="mw-heading mw-heading2"><h2 id="Algebraic_form">Algebraic form</h2></div>

In <a href="Vector_notation" title="Vector notation">vector notation</a>, a plane can be expressed as the set of points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }">

</mrow>

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

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</math><img src="./dd73e3862cb92b016721b8c492eadb4e8a577527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }" loading="lazy"> for which

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {p} -\mathbf {p_{0}} )\cdot \mathbf {n} =0}</annotation>

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</math><img src="./ab1a56a9f588e49135ad8f9d492bac96606abff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.223ex; height:2.843ex;" alt="{\displaystyle (\mathbf {p} -\mathbf {p_{0}} )\cdot \mathbf {n} =0}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }">

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

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</math><img src="./4a720c341f39f52fd96028dab83edd34d400be46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" loading="lazy"> is a <a href="Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> to the plane and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p_{0}} }">

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

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</math><img src="./ca639d9feafabd7c3de9a9721ef714774ea380ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.663ex; height:2.009ex;" alt="{\displaystyle \mathbf {p_{0}} }" loading="lazy"> is a point on the plane. (The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot \mathbf {b} }">

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

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</math><img src="./494aed3b5e94f1c0ee071debc707d2700c0e0390.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.464ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} \cdot \mathbf {b} }" loading="lazy"> denotes the <a href="Dot_product" title="Dot product">dot product</a> of the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }">

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

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</math><img src="./1a957216653a9ee0d0133dcefd13fb75e36b8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

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

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</math><img src="./13ebf4628a1adf07133a6009e4a78bdd990c6eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {b} }" loading="lazy">.)

The vector equation for a line is

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\mathbf {l_{0}} +\mathbf {l} \ d\quad d\in \mathbb {R} }</annotation>

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</math><img src="./920bf9e4d8271bcf9fc2c3e225cde892e7c6620a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.939ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =\mathbf {l_{0}} +\mathbf {l} \ d\quad d\in \mathbb {R} }" loading="lazy"></dd></dl>

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

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</math><img src="./dca0b04733c4e44533df8a7eb12145d74cdbefef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.742ex; height:2.176ex;" alt="{\displaystyle \mathbf {l} }" loading="lazy"> is a unit vector in the direction of the line, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l_{0}} }">

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

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</math><img src="./f92ce73822637973e6bfb3b1656c944d30202947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.919ex; height:2.509ex;" alt="{\displaystyle \mathbf {l_{0}} }" loading="lazy"> is a point on the line, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> is a scalar in the <a href="Real_number" title="Real number">real number</a> domain. Substituting the equation for the line into the equation for the plane gives

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<annotation encoding="application/x-tex">{\displaystyle ((\mathbf {l_{0}} +\mathbf {l} \ d)-\mathbf {p_{0}} )\cdot \mathbf {n} =0.}</annotation>

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</math><img src="./ec86041f93512f8831f51996dd9d125d514b4e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.492ex; height:2.843ex;" alt="{\displaystyle ((\mathbf {l_{0}} +\mathbf {l} \ d)-\mathbf {p_{0}} )\cdot \mathbf {n} =0.}" loading="lazy"></dd></dl>

Expanding gives

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {l} \cdot \mathbf {n} )\ d+(\mathbf {l_{0}} -\mathbf {p_{0}} )\cdot \mathbf {n} =0.}</annotation>

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</math><img src="./f7e12a2aa7c4100316459d3f0c07c2ac4b811fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.657ex; height:2.843ex;" alt="{\displaystyle (\mathbf {l} \cdot \mathbf {n} )\ d+(\mathbf {l_{0}} -\mathbf {p_{0}} )\cdot \mathbf {n} =0.}" loading="lazy"></dd></dl>

And solving for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d={(\mathbf {p_{0}} -\mathbf {l_{0}} )\cdot \mathbf {n} \over \mathbf {l} \cdot \mathbf {n} }.}</annotation>

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</math><img src="./74828a6d17ad34b2bbb6f3760bce6b436a88bafb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.194ex; height:5.676ex;" alt="{\displaystyle d={(\mathbf {p_{0}} -\mathbf {l_{0}} )\cdot \mathbf {n} \over \mathbf {l} \cdot \mathbf {n} }.}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} \cdot \mathbf {n} =0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} \cdot \mathbf {n} =0}</annotation>

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</math><img src="./826c3d32c2d6c6728e1b6b6a606ec00e5f98dca1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.167ex; height:2.176ex;" alt="{\displaystyle \mathbf {l} \cdot \mathbf {n} =0}" loading="lazy"> then the line and plane are parallel. There will be two cases: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {p_{0}} -\mathbf {l_{0}} )\cdot \mathbf {n} =0}">

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {p_{0}} -\mathbf {l_{0}} )\cdot \mathbf {n} =0}</annotation>

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</math><img src="./aba58b89ae8f93d2cf24435379b47ae4a7f7f88a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.657ex; height:2.843ex;" alt="{\displaystyle (\mathbf {p_{0}} -\mathbf {l_{0}} )\cdot \mathbf {n} =0}" loading="lazy"> then the line is contained in the plane, that is, the line intersects the plane at each point of the line. Otherwise, the line and plane have no intersection.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} \cdot \mathbf {n} \neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} \cdot \mathbf {n} \neq 0}</annotation>

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</math><img src="./1bd5a1ba48c18488cdb291a6d3d6596ec494e237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.167ex; height:2.676ex;" alt="{\displaystyle \mathbf {l} \cdot \mathbf {n} \neq 0}" loading="lazy"> there is a single point of intersection. The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> can be calculated and the point of intersection, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }">

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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\mathbf {l_{0}} +\mathbf {l} \ d}</annotation>

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</math><img src="./a99d36b75b778453f9fd3c4c62fb6f3f1925f260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.882ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =\mathbf {l_{0}} +\mathbf {l} \ d}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Parametric_form">Parametric form</h2></div>

A line is described by all points that are a given direction from a point. A general point on a line passing through points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{a}=(x_{a},y_{a},z_{a})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{a}=(x_{a},y_{a},z_{a})}</annotation>

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</math><img src="./e32a6c2d7eb4e77670d865b95723e1572c3f603d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.675ex; height:2.843ex;" alt="{\displaystyle \mathbf {l} _{a}=(x_{a},y_{a},z_{a})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{b}=(x_{b},y_{b},z_{b})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{b}=(x_{b},y_{b},z_{b})}</annotation>

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</math><img src="./31e60274e94a2b9eaba061d6d347f841a61f5bd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.018ex; height:2.843ex;" alt="{\displaystyle \mathbf {l} _{b}=(x_{b},y_{b},z_{b})}" loading="lazy"> can be represented as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t,\quad t\in \mathbb {R} ,}</annotation>

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</math><img src="./08ed8a5b45accc9717f7857f33d009273aed4af2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.435ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t,\quad t\in \mathbb {R} ,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{ab}=\mathbf {l} _{b}-\mathbf {l} _{a}}</annotation>

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</math><img src="./08e1ab3e9234b16c017138c5c7a686dd5f1dc9db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.012ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{ab}=\mathbf {l} _{b}-\mathbf {l} _{a}}" loading="lazy"> is the vector pointing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{a}}">

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

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</math><img src="./24c63bac32ab4357b8e058003cf22074e9441187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.844ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{a}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{b}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{b}}</annotation>

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Similarly a general point on a plane determined by the triangle defined by the points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{0}=(x_{0},y_{0},z_{0})}">

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

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</math><img src="./86353058967f318f37cd37c9932762742accd2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.228ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} _{0}=(x_{0},y_{0},z_{0})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{1}=(x_{1},y_{1},z_{1})}">

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

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</math><img src="./1486b7389012f41db31fdb55c0c0bb3e67ba68aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.228ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} _{1}=(x_{1},y_{1},z_{1})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{2}=(x_{2},y_{2},z_{2})}">

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

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</math><img src="./75c11af2b1e0706cf0faf479edc90d058749594c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.228ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} _{2}=(x_{2},y_{2},z_{2})}" loading="lazy"> can be represented as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{0}+\mathbf {p} _{01}u+\mathbf {p} _{02}v,\quad u,v\in \mathbb {R} ,}</annotation>

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</math><img src="./faa06cc3bf3130dc9cc676b38205c34bbd794c5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.414ex; height:2.676ex;" alt="{\displaystyle \mathbf {p} _{0}+\mathbf {p} _{01}u+\mathbf {p} _{02}v,\quad u,v\in \mathbb {R} ,}" loading="lazy"></dd></dl>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{01}=\mathbf {p} _{1}-\mathbf {p} _{0}}</annotation>

</semantics>

</math><img src="./e21bc01ee1ae2a51c3c1d3561e1957a1b0b0d121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.379ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} _{01}=\mathbf {p} _{1}-\mathbf {p} _{0}}" loading="lazy"> is the vector pointing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ae6eb9d63094845ac7406038dc1ed8803f6ef575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{0}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{1}}</annotation>

</semantics>

</math><img src="./279e30b6f8eba8d6ff75f984f9393eebc2ea6d6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{1}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{02}=\mathbf {p} _{2}-\mathbf {p} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{02}=\mathbf {p} _{2}-\mathbf {p} _{0}}</annotation>

</semantics>

</math><img src="./a9324b8f2ebcba497d7af49e4f22f8cfbbc868c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.379ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} _{02}=\mathbf {p} _{2}-\mathbf {p} _{0}}" loading="lazy"> is the vector pointing from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{2}}</annotation>

</semantics>

</math><img src="./0503cf42886fcd7909a39b737ba760b0249b3ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{2}}" loading="lazy">.

The point at which the line intersects the plane is therefore described by setting the point on the line equal to the point on the plane, giving the parametric equation:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t=\mathbf {p} _{0}+\mathbf {p} _{01}u+\mathbf {p} _{02}v.}</annotation>

</semantics>

</math><img src="./7c2ff40b33ab73280cc733b4e82cdf156a6a4f27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.219ex; height:2.676ex;" alt="{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t=\mathbf {p} _{0}+\mathbf {p} _{01}u+\mathbf {p} _{02}v.}" loading="lazy"></dd></dl>

This can be rewritten as

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{a}-\mathbf {p} _{0}=-\mathbf {l} _{ab}t+\mathbf {p} _{01}u+\mathbf {p} _{02}v,}</annotation>

</semantics>

</math><img src="./839b29434f672dc5406b1799da8326121a39e971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.027ex; height:2.676ex;" alt="{\displaystyle \mathbf {l} _{a}-\mathbf {p} _{0}=-\mathbf {l} _{ab}t+\mathbf {p} _{01}u+\mathbf {p} _{02}v,}" loading="lazy"></dd></dl>

which can be expressed in matrix form as

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}}={\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}}{\begin{bmatrix}t\\u\\v\end{bmatrix}},}</annotation>

</semantics>

</math><img src="./3701d7c9c4c720ef96fa3fabfa96837e3fc2b403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:35.967ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}}={\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}}{\begin{bmatrix}t\\u\\v\end{bmatrix}},}" loading="lazy"></dd></dl>

where the vectors are written as column vectors.

This produces a <a href="System_of_linear_equations" title="System of linear equations">system of linear equations</a> which can be solved for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">. If the solution satisfies the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [0,1],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in [0,1],}</annotation>

</semantics>

</math><img src="./c327d1a07b1551a3ec5fc7bda0996d4ed770e462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.98ex; height:2.843ex;" alt="{\displaystyle t\in [0,1],}" loading="lazy">, then the intersection point is on the line segment between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{a}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./24c63bac32ab4357b8e058003cf22074e9441187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.844ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{a}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{b}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{b}}</annotation>

</semantics>

</math><img src="./3bcbded2eb4a4a04f565dd5c1c44f4c4a569492d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.68ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{b}}" loading="lazy">, otherwise it is elsewhere on the line. Likewise, if the solution satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in [0,1],}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\in [0,1],}</annotation>

</semantics>

</math><img src="./c7f75527bca006732f85c0a50f7e05369a95de37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.631ex; height:2.843ex;" alt="{\displaystyle u,v\in [0,1],}" loading="lazy">, then the intersection point is in the <a href="Parallelogram" title="Parallelogram">parallelogram</a> formed by the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ae6eb9d63094845ac7406038dc1ed8803f6ef575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{0}}" loading="lazy"> and vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{01}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{01}}</annotation>

</semantics>

</math><img src="./eaf699ff09e85d1e97dae17ae664d802cb62c490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.362ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{01}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{02}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{02}}</annotation>

</semantics>

</math><img src="./c405dc68f4236eab73d1cd7ba2e8176b399adc29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.362ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{02}}" loading="lazy">. If the solution additionally satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u+v)\leq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u+v)\leq 1}</annotation>

</semantics>

</math><img src="./3d7de14c8be152dd9c799bd4cd3401913d696b4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (u+v)\leq 1}" loading="lazy">, then the intersection point lies in the triangle formed by the three points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ae6eb9d63094845ac7406038dc1ed8803f6ef575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{0}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{1}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{1}}</annotation>

</semantics>

</math><img src="./279e30b6f8eba8d6ff75f984f9393eebc2ea6d6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.54ex; height:2.176ex;" alt="{\displaystyle \mathbf {p} _{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} _{2}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {p} _{2}}</annotation>

</semantics>

The determinant of the matrix can be calculated as

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det({\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}})=-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02}).}</annotation>

</semantics>

</math><img src="./50e0e2f20d9aab7a2e6322d2ea71ae36bb52e21b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.964ex; height:2.843ex;" alt="{\displaystyle \det({\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}})=-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02}).}" loading="lazy"></dd></dl>

If the determinant is zero, then there is no unique solution; the line is either in the plane or parallel to it.

If a unique solution exists (determinant is not 0), then it can be found by <a href="Matrix_inversion" class="mw-redirect" title="Matrix inversion">inverting</a> the matrix and rearranging:

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}}^{-1}{\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}},}</annotation>

</semantics>

</math><img src="./d96252e964490fb3a9055be3ea692923ae6bd9f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.3ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\begin{bmatrix}-\mathbf {l} _{ab}&\mathbf {p} _{01}&\mathbf {p} _{02}\end{bmatrix}}^{-1}{\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}},}" loading="lazy"></dd></dl>

which expands to

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\frac {1}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}{\begin{bmatrix}{(\mathbf {p} _{01}\times \mathbf {p} _{02})}^{\mathrm {T} }\\{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}^{\mathrm {T} }\\{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}^{\mathrm {T} }\end{bmatrix}}{\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}}}</annotation>

</semantics>

</math><img src="./be1c29d0400257950334ff9e2f65a17f323df23d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:53.434ex; height:10.843ex;" alt="{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\frac {1}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}{\begin{bmatrix}{(\mathbf {p} _{01}\times \mathbf {p} _{02})}^{\mathrm {T} }\\{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}^{\mathrm {T} }\\{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}^{\mathrm {T} }\end{bmatrix}}{\begin{bmatrix}\mathbf {l} _{a}-\mathbf {p} _{0}\end{bmatrix}}}" loading="lazy"></dd></dl>

and then to

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

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<msub>

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</mtd>

</mtr>

<mtr>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\frac {1}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}{\begin{bmatrix}{(\mathbf {p} _{01}\times \mathbf {p} _{02})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\\{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\\{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\end{bmatrix}},}</annotation>

</semantics>

</math><img src="./71925c0b78c46579a41f4c7c3bc37a96d8ab1743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:54.106ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}t\\u\\v\end{bmatrix}}={\frac {1}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}{\begin{bmatrix}{(\mathbf {p} _{01}\times \mathbf {p} _{02})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\\{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\\{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})\end{bmatrix}},}" loading="lazy"></dd></dl>

thus giving the solutions:

<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t={\frac {{(\mathbf {p} _{01}\times \mathbf {p} _{02})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}}</annotation>

</semantics>

</math><img src="./d15c7464a9a3c25c55a13d964fe37d38a3648f94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.859ex; height:6.509ex;" alt="{\displaystyle t={\frac {{(\mathbf {p} _{01}\times \mathbf {p} _{02})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}}" loading="lazy"></dd>

<mfrac>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle u={\frac {{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}}</annotation>

</semantics>

</math><img src="./3ece713463bcca14445cb33405e3d706a6c1c570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.345ex; height:6.509ex;" alt="{\displaystyle u={\frac {{(\mathbf {p} _{02}\times -\mathbf {l} _{ab})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}}" loading="lazy"></dd>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle v={\frac {{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}.}</annotation>

</semantics>

</math><img src="./e5c50a94982aa66a0c835784395453e42f200a4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.79ex; height:6.509ex;" alt="{\displaystyle v={\frac {{(-\mathbf {l} _{ab}\times \mathbf {p} _{01})}\cdot (\mathbf {l} _{a}-\mathbf {p} _{0})}{-\mathbf {l} _{ab}\cdot (\mathbf {p} _{01}\times \mathbf {p} _{02})}}.}" loading="lazy"></dd></dl>

The point of intersection is then equal to

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t}</annotation>

</semantics>

</math><img src="./38997b16f5cdfec3abcc563b1733d30b4026f013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.073ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{a}+\mathbf {l} _{ab}t}" loading="lazy"></dd></dl>

In the <a href="Ray_tracing_(graphics)" title="Ray tracing (graphics)">ray tracing</a> method of <a href="Computer_graphics" title="Computer graphics">computer graphics</a> a surface can be represented as a set of pieces of planes. The intersection of a ray of light with each plane is used to produce an image of the surface. In vision-based <a href="3D_reconstruction" title="3D reconstruction">3D reconstruction</a>, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera.

The algorithm can be generalised to cover intersection with other planar figures, in particular, the <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>.

<ul><li><a href="Pl%C3%BCcker_coordinates#Plane-line_meet" title="Plücker coordinates">Plücker coordinates#Plane-line meet</a> calculating the intersection when the line is expressed by Plücker coordinates.</li>

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

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

<ul><li><a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/1%3A_Vectors_in_Space/Intersection_of_a_Line_and_a_Plane">Intersection of a Line and a Plane</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-12-24" href="https://en.wikipedia.org/wiki/?title=Line%E2%80%93plane_intersection&oldid=1265020552">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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