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Line–sphere intersection
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<h2 id="calculation-using-vectors-in-3d">Calculation using vectors in 3D</h2> <p>In <a href="/facts/Vector_notation/jfavhXKy">vector notation</a>, the equations are as follows: </p><p>Equation for a <a href="/facts/Sphere/jywYLNM2">sphere</a> </p> ‖ x − c ‖ 2 = r 2 {\displaystyle \left\Vert \mathbf {x} -\mathbf {c} \right\Vert ^{2}=r^{2}} <ul><li> x {\displaystyle \mathbf {x} } : points on the sphere</li> <li> c {\displaystyle \mathbf {c} } : center point</li> <li> r {\displaystyle r} : radius of the sphere</li></ul> <p>Equation for a line starting at o {\displaystyle \mathbf {o} } </p> x = o + d u {\displaystyle \mathbf {x} =\mathbf {o} +d\mathbf {u} } <ul><li> x {\displaystyle \mathbf {x} } : points on the line</li> <li> o {\displaystyle \mathbf {o} } : origin of the line</li> <li> d {\displaystyle d} : distance from the origin of the line</li> <li> u {\displaystyle \mathbf {u} } : direction of line (a non-zero vector)</li></ul> <p>Searching for points that are on the line and on the sphere means combining the equations and solving for d {\displaystyle d} , involving the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of vectors: </p> Equations combined ‖ o + d u − c ‖ 2 = r 2 ⇔ ( o + d u − c ) ⋅ ( o + d u − c ) = r 2 {\displaystyle \left\Vert \mathbf {o} +d\mathbf {u} -\mathbf {c} \right\Vert ^{2}=r^{2}\Leftrightarrow (\mathbf {o} +d\mathbf {u} -\mathbf {c} )\cdot (\mathbf {o} +d\mathbf {u} -\mathbf {c} )=r^{2}} Expanded and rearranged: d 2 ( u ⋅ u ) + 2 d [ u ⋅ ( o − c ) ] + ( o − c ) ⋅ ( o − c ) − r 2 = 0 {\displaystyle d^{2}(\mathbf {u} \cdot \mathbf {u} )+2d[\mathbf {u} \cdot (\mathbf {o} -\mathbf {c} )]+(\mathbf {o} -\mathbf {c} )\cdot (\mathbf {o} -\mathbf {c} )-r^{2}=0} The form of a <a href="/facts/Quadratic_formula/j5ftMowG">quadratic formula</a> is now observable. (This quadratic equation is an instance of Joachimsthal's equation.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>) a d 2 + b d + c = 0 {\displaystyle ad^{2}+bd+c=0} where <ul><li> a = u ⋅ u = ‖ u ‖ 2 {\displaystyle a=\mathbf {u} \cdot \mathbf {u} =\left\Vert \mathbf {u} \right\Vert ^{2}} </li> <li> b = 2 [ u ⋅ ( o − c ) ] {\displaystyle b=2[\mathbf {u} \cdot (\mathbf {o} -\mathbf {c} )]} </li> <li> c = ( o − c ) ⋅ ( o − c ) − r 2 = ‖ o − c ‖ 2 − r 2 {\displaystyle c=(\mathbf {o} -\mathbf {c} )\cdot (\mathbf {o} -\mathbf {c} )-r^{2}=\left\Vert \mathbf {o} -\mathbf {c} \right\Vert ^{2}-r^{2}} </li></ul> Simplified d = − 2 [ u ⋅ ( o − c ) ] ± ( 2 [ u ⋅ ( o − c ) ] ) 2 − 4 ‖ u ‖ 2 ( ‖ o − c ‖ 2 − r 2 ) 2 ‖ u ‖ 2 {\displaystyle d={\frac {-2[\mathbf {u} \cdot (\mathbf {o} -\mathbf {c} )]\pm {\sqrt {(2[\mathbf {u} \cdot (\mathbf {o} -\mathbf {c} )])^{2}-4\left\Vert \mathbf {u} \right\Vert ^{2}(\left\Vert \mathbf {o} -\mathbf {c} \right\Vert ^{2}-r^{2})}}}{2\left\Vert \mathbf {u} \right\Vert ^{2}}}} Note that in the specific case where u {\displaystyle \mathbf {u} } is a unit vector, and thus ‖ u ‖ 2 = 1 {\displaystyle \left\Vert \mathbf {u} \right\Vert ^{2}=1} , we can simplify this further to (writing u ^ {\displaystyle {\hat {\mathbf {u} }}} instead of u {\displaystyle \mathbf {u} } to indicate a unit vector): ∇ = [ u ^ ⋅ ( o − c ) ] 2 − ( ‖ o − c ‖ 2 − r 2 ) {\displaystyle \nabla =[{\hat {\mathbf {u} }}\cdot (\mathbf {o} -\mathbf {c} )]^{2}-(\left\Vert \mathbf {o} -\mathbf {c} \right\Vert ^{2}-r^{2})} d = − [ u ^ ⋅ ( o − c ) ] ± ∇ {\displaystyle d=-[{\hat {\mathbf {u} }}\cdot (\mathbf {o} -\mathbf {c} )]\pm {\sqrt {\nabla }}} <ul><li>If ∇ < 0 {\displaystyle \nabla <0} , then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1).</li> <li>If ∇ = 0 {\displaystyle \nabla =0} , then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2).</li> <li>If ∇ > 0 {\displaystyle \nabla >0} , two solutions exist, and thus the line touches the sphere in two points (case 3).</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Intersection_(geometry)/RwH57yPq">Intersection (geometry) § A line and a circle</a></li> <li><a href="/facts/Analytic_geometry/3qWP0kM0">Analytic geometry</a></li> <li><a href="/facts/Line%25E2%2580%2593plane_intersection/N52PmX7z">Line–plane intersection</a></li> <li><a href="/facts/Plane%25E2%2580%2593plane_intersection/tAUtRFcn">Plane–plane intersection</a></li> <li><a href="/facts/Plane%25E2%2580%2593sphere_intersection/a3W08Qrg">Plane–sphere intersection</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Eberly, David H. (2006). 3D game engine design: a practical approach to real-time computer graphics, 2nd edition. Morgan Kaufmann. p. 698. ISBN 0-12-229063-1. <a href="0-12-229063-1" target="_blank">0-12-229063-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Joachimsthal's Equation". <a href="http://mathworld.wolfram.com/JoachimsthalsEquation.html" target="_blank">http://mathworld.wolfram.com/JoachimsthalsEquation.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>