A question about polygons

1. A question about polygons Devising a computational method for determining if a point lies in a plane convex polygon

2. Problem background • Recall our ‘globe.cpp’ ray-tracing demo • It depicted a scene showing two objects • One of the objects was a square tabletop • Its edges ran parallel to x- and z-axes • That fact made it easy to determine if the spot where a ray hit the tabletop fell inside or outside of the square’s boundary edges • We were lucky!

3. Alternative geometry? • What if we wanted to depict our globe as resting on a tabletop that wasn’t a nicely aligned square? For example, could we show a tabletop that was a hexagon? • The only change is that the edges of this six-sided tabletop can’t all be lined up with the coordinate system axes • We need a new approach to determining if a ray hits a spot that’s on our tabletop

4. A simpler problem • How can we tell if a point lies in a triangle? c q p b a Point p lies inside triangle Δabc Point q lies outside triangle Δabc

5. Triangle algorithm • Draw vectors from a to b and from a to c • We can regard these vectors as the axes for a “skewed” coordinate system • Then every point in the triangle’s plane would have a unique pair of coordinates • We can compute those coordinates using Cramer’s Rule (from linear algebra)

6. The algorithm idea c ap = c1*ab + c2*ac p b a c1 = det( ap, ac )/det( ab, ac ) c2 = det( ab, ap )/det( ab, ac )

7. Cartesian Analogy y-axis p = (c1,c2) x-axis x + y = 1 p lies outside the triangle if c1 < 0 or c2 < 0 or c1+c2 > 1

8. Determinant function: 2x2 typedef float scalar_t; typedef struct { scalar_t x, y; } vector_t; scalar_t det( vector_t a, vector_t b ) { return a.x * b.y – b.x * a.y; }

9. Constructing a regular hexagon theta = 2*PI / 6 ( cos(2*theta), sin(2*theta) ) ( cos(1*theta), sin(1*theta) ) ( cos(3*theta), sin(3*theta) ) ( cos(0*theta), sin(0*theta) ) ( cos(5*theta), sin(5*theta) ) ( cos(4*theta), sin(4*theta) )

10. Subdividing the hexagon A point p lies outside the hexagon -- unless it lies inside one of these four sub-triangles

11. Same approach for n-gons • Demo program ‘hexagon.cpp’ illustrates the use of our just-described algorithm • Every convex polygon can be subdivided into triangles, so the same ideas can be applied to any regular n-sided polygon • Exercise: modify the demo-program so it draws an octagon, a pentagon, a septagon

12. Extension to a tetrahedron • A tetrahedron is a 3D analog of a triangle • It has 4 vertices, located in space (but not all vertices can lie in the same plane) • Each face of a tetrahedron is a triangle c b o a

13. Cramer’s Rule in 3D typedef float scalar_t; typedef struct { scalar_t x, y, z; } vector_t; scalar_t det( vector_t a, vector_t b, vector_t c ) { scalar_t sum = 0.0; sum += a.x * b.y * c.z – a.x * b.z * c.y; sum += a.y * b.z * c.x – a.y * b.x * c.z; sum += a.z * b.x * c.y – a.z * b.y * c.x; return sum; }

14. Is point in tetrahedron? • Let o, a, b, c be vertices of a tetrahedron • Form the three vectors oa, ob, oc and regard them as the coordinate axes in a “skewed” 3D coordinate system • Then any point p in space has a unique triple of coordinates: op = c1*oa + c2*ob + c3*oc • These three coordinates can be computed using Cramer’s Rule

15. Details of Cramer Rule c1 = det( op, ob, oc )/det( oa, ob, oc ) c2 = det( oa, op, oc )/det( oa, ob, oc ) c3 = det( oa, ob, op )/det( oa, ob, oc ) Point p lies inside the tetrahedron – unless c1 < 0 or c2 < 0 or c3 < 0 or c1 + c2 + c3 > 1

16. Convex polyhedron • Just as a convex polygon can be divided into subtriangles, any convex polyhedron can be divided into several tetrahedrons • We can tell if a point lies in the polyhedron by testing to see it lies in one of the solid’s tetrahedral parts • An example: the regular dodecahedron can be raytraced by using these ideas!