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Math: clarify that a perp-dot product is used here as well.

pull/603/head
Vladimír Vondruš 4 years ago
parent
commit
7ac7e423d9
  1. 12
      src/Magnum/Math/Intersection.h

12
src/Magnum/Math/Intersection.h

@ -111,14 +111,16 @@ Returns intersection point positions @f$ t @f$, @f$ u @f$ on both lines:
The two lines intersect if @f$ t @f$ and @f$ u @f$ exist such that: @f[
\boldsymbol{p} + t \boldsymbol{r} = \boldsymbol{q} + u \boldsymbol{s}
@f]
Crossing both sides with @f$ \boldsymbol{s} @f$, distributing the cross product
and eliminating @f$ \boldsymbol{s} \times \boldsymbol{s} = 0 @f$, then solving
for @f$ t @f$ and similarly for @f$ u @f$: @f[
Crossing both sides with @f$ \boldsymbol{s} @f$
(a @ref cross(const Vector2<T>&, const Vector2<T>&) "perp-dot product"),
distributing the cross product and eliminating
@f$ \boldsymbol{s} \times \boldsymbol{s} = 0 @f$, then solving for @f$ t @f$
and similarly for @f$ u @f$: @f[
\begin{array}{rcl}
(\boldsymbol{p} + t \boldsymbol{r}) \times \boldsymbol{s} & = & (\boldsymbol{q} + u \boldsymbol{s}) \times \boldsymbol{s} \\
t (\boldsymbol{r} \times \boldsymbol{s}) & = & (\boldsymbol{q} - \boldsymbol{p}) \times \boldsymbol{s} \\
t & = & \cfrac{(\boldsymbol{q} - \boldsymbol{p}) \times \boldsymbol{s}}{\boldsymbol{r} \times \boldsymbol{s}} \\
u & = & \cfrac{(\boldsymbol{q} - \boldsymbol{p}) \times \boldsymbol{r}}{\boldsymbol{r} \times \boldsymbol{s}}
t & = & \cfrac{(\boldsymbol{q} - \boldsymbol{p}) \times \boldsymbol{s}}{\boldsymbol{r} \times \boldsymbol{s}} = \cfrac{(\boldsymbol{q} - \boldsymbol{p})_\bot \cdot \boldsymbol{s}}{\boldsymbol{r}_\bot \cdot \boldsymbol{s}} \\
u & = & \cfrac{(\boldsymbol{q} - \boldsymbol{p}) \times \boldsymbol{r}}{\boldsymbol{r} \times \boldsymbol{s}} = \cfrac{(\boldsymbol{q} - \boldsymbol{p})_\bot \cdot \boldsymbol{r}}{\boldsymbol{r}_\bot \cdot \boldsymbol{s}}
\end{array}
@f]

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