#ifndef Magnum_Math_Geometry_Intersection_h #define Magnum_Math_Geometry_Intersection_h /* This file is part of Magnum. Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018 Vladimír Vondruš Copyright © 2016 Jonathan Hale Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */ /** @file * @brief Namespace @ref Magnum::Math::Geometry::Intersection */ #include "Magnum/Math/Frustum.h" #include "Magnum/Math/Geometry/Distance.h" #include "Magnum/Math/Range.h" #include "Magnum/Math/Vector3.h" namespace Magnum { namespace Math { namespace Geometry { namespace Intersection { /** @brief Intersection of two line segments in 2D @param p Starting point of first line segment @param r Direction of first line segment @param q Starting point of second line segment @param s Direction of second line segment Returns intersection point positions @f$ t @f$, @f$ u @f$ on both lines, NaN if the lines are collinear or infinity if they are parallel. Intersection point can be then calculated with @f$ \boldsymbol{p} + t \boldsymbol{r} @f$ or @f$ \boldsymbol{q} + u \boldsymbol{s} @f$. If @f$ t @f$ is in range @f$ [ 0 ; 1 ] @f$, the intersection is inside the line segment defined by @f$ \boldsymbol{p} @f$ and @f$ \boldsymbol{p} + \boldsymbol{r} @f$, if @f$ u @f$ is in range @f$ [ 0 ; 1 ] @f$, the intersection is inside the line segment defined by @f$ \boldsymbol{q} @f$ and @f$ \boldsymbol{q} + \boldsymbol{s} @f$. The two lines intersect if @f$ 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$: @f[ \begin{array}{rcl} (\boldsymbol p + t \boldsymbol r) \times s & = & (\boldsymbol q + u \boldsymbol s) \times s \\ t (\boldsymbol r \times s) & = & (\boldsymbol q - \boldsymbol p) \times s \\ t & = & \cfrac{(\boldsymbol q - \boldsymbol p) \times s}{\boldsymbol r \times \boldsymbol s} \\ u & = & \cfrac{(\boldsymbol q - \boldsymbol p) \times r}{\boldsymbol r \times \boldsymbol s} \end{array} @f] See also @ref lineSegmentLine() which calculates only @f$ t @f$, useful if you don't need to test that the intersection lies inside line segment defined by @f$ \boldsymbol{q} @f$ and @f$ \boldsymbol{q} + \boldsymbol{s} @f$. */ template inline std::pair lineSegmentLineSegment(const Vector2& p, const Vector2& r, const Vector2& q, const Vector2& s) { const Vector2 qp = q - p; const T rs = cross(r, s); return {cross(qp, s)/rs, cross(qp, r)/rs}; } /** @brief Intersection of line segment and line in 2D @param p Starting point of first line segment @param r Direction of first line segment @param q Starting point of second line @param s Direction of second line Returns intersection point position @f$ t @f$ on first line, NaN if the lines are collinear or infinity if they are parallel. Intersection point can be then calculated with @f$ \boldsymbol{p} + t \boldsymbol{r} @f$. If returned value is in range @f$ [ 0 ; 1 ] @f$, the intersection is inside the line segment defined by @f$ \boldsymbol{p} @f$ and @f$ \boldsymbol{p} + \boldsymbol{r} @f$. Unlike @ref lineSegmentLineSegment() calculates only @f$ t @f$. */ template inline T lineSegmentLine(const Vector2& p, const Vector2& r, const Vector2& q, const Vector2& s) { return cross(q - p, s)/cross(r, s); } /** @brief Intersection of a plane and line @param planePosition Plane position @param planeNormal Plane normal @param p Starting point of the line @param r Direction of the line Returns intersection point position @f$ t @f$ on the line, NaN if the line lies on the plane or infinity if the intersection doesn't exist. Intersection point can be then calculated from with @f$ \boldsymbol{p} + t \boldsymbol{r} @f$. If returned value is in range @f$ [ 0 ; 1 ] @f$, the intersection is inside the line segment defined by @f$ \boldsymbol{p} @f$ and @f$ \boldsymbol{r} @f$. First the parameter @f$ f @f$ of parametric equation of the plane is calculated from plane normal @f$ \boldsymbol{n} @f$ and plane position: @f[ \begin{pmatrix} n_0 \\ n_1 \\ n_2 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} - f = 0 @f] Using plane normal @f$ \boldsymbol{n} @f$, parameter @f$ f @f$ and line defined by @f$ \boldsymbol{p} @f$ and @f$ \boldsymbol{r} @f$, value of @f$ t @f$ is calculated and returned. @f[ \begin{array}{rcl} f & = & \boldsymbol n \cdot (\boldsymbol p + t \boldsymbol r) \\ \Rightarrow t & = & \cfrac{f - \boldsymbol n \cdot \boldsymbol p}{\boldsymbol n \cdot \boldsymbol r} \end{array} @f] */ template inline T planeLine(const Vector3& planePosition, const Vector3& planeNormal, const Vector3& p, const Vector3& r) { const T f = dot(planePosition, planeNormal); return (f - dot(planeNormal, p))/dot(planeNormal, r); } /** @brief Intersection of a point and a camera frustum @param point Point @param frustum Frustum planes with normals pointing outwards Returns `true` if the point is on or inside the frustum. Checks for each plane of the frustum whether the point is behind the plane (the points distance from the plane is negative) using @ref Distance::pointPlaneScaled(). */ template bool pointFrustum(const Vector3& point, const Frustum& frustum); /** @brief Intersection of an axis-aligned box and a camera frustum @param box Axis-aligned box @param frustum Frustum planes with normals pointing outwards Returns `true` if the box intersects with the camera frustum. Counts for each plane of the frustum how many points of the box lie in front of the plane (outside of the frustum). If none, the box must lie entirely outside of the frustum and there is no intersection. Else, the box is considered as intersecting, even if it is merely corners of the box overlapping with corners of the frustum, since checking the corners is less efficient. */ template bool boxFrustum(const Range3D& box, const Frustum& frustum); template bool pointFrustum(const Vector3& point, const Frustum& frustum) { for(const Vector4& plane: frustum.planes()) { /* The point is in front of one of the frustum planes (normals point outwards) */ if(Distance::pointPlaneScaled(point, plane) < T(0)) return false; } return true; } template bool boxFrustum(const Range3D& box, const Frustum& frustum) { for(const Vector4& plane: frustum.planes()) { bool cornerHit = 0; for(UnsignedByte c = 0; c != 8; ++c) { const Vector3 corner = Math::lerp(box.min(), box.max(), Math::BoolVector<3>{c}); if(Distance::pointPlaneScaled(corner, plane) >= T(0)) { cornerHit = true; break; } } /* All corners are outside this plane */ if(!cornerHit) return false; } /** @todo potentially check corners here to avoid false positives */ return true; } }}}} #endif