Object Intersection
CSE 681 Object Representation
Implicit forms F(x,y,z) = 0 ttitesting
Explicit forms Analytic form x = F(y,z) generating
Parametric form (x,y,z) = P(t)
CSE 681 Ray-Object Intersection
Implicit forms F(x,y,z) = 0
Ray: P(t) = (x,y,z) = source + t*direction = s + t*v
Solve for t: F(P(t)) = 0
CSE 681 Ray-Sphere Intersection
Implicit form for sphere at origin of radius 1 F(x, y, z) = x2 + y 2 + z 2 −1 = 0
Ray: P(t) = (x, y, z) = s + tv = (sx + tvx , sy + tvy , sz + tvz )
Solve: … 2 2 2 F(P(t)) = (sx + tvx ) + (sy + tvy ) + (sz + tvz ) −1 = 0 2 2 2 2 2 2 2 = sx + sy + sz + 2t(sxvx + syvy + szvz ) + t (vx + vy + vz ) −1 = 0
Use quadratic equation…
CSE 681 Ray-Sphere Intersection
At2 + Bt + C = 0 A = |v|2 B=2sB = 2s· v C = |s|2 -1
− B ± B2 − 4AC t = 2A
B2-4AC < 0 => no intersection = 0 => just grazes > 0 => t wo hit s
CSE 681 Axis-Aligned Cuboid (rectangular solid, rectangular parallelpiped) RtiRay equation P(t) = s + tv Planar equations
Solve for intersections with planes Y = y2
tx1 = (x1 - sx)/vx Z = z2 t = (x2 - s )/v X = x1 X = x2 x2 x x Z = z1 ty1 = (y1 - sx)/vx Y = y1 …
CSE 681 For each iiintersection, note Rectangle whether entering or leavinggq square side of plane: HOW?
Ray 2 Ray is inside after last entering and Ray 1 before first exiting
CSE 681 Ray-Plane for arbitraryyp plane Generalize from axis-aligned planes to any plane:
V V2 ax + by + cz + d = 0 1
n ·P = -d
d -n V4
V3
CSE 681 Normal Vector
Given ordered sequence of points defining a polygon how do you find a normal vector for the plane?
Note: 2 normal vectors to a plane, colinear and one is the negation of the other
Ordered: e.g., clockwise when viewed from the front of the face
Riggpht hand v. left hand space
CSE 681 Normal Vector V1 V2
V3 n = (V1 -V2) x (V3 -V2)
V5 V4 V2
V1 V1 V3 V2 V3
V5 V 4 V5
CSE 681 V4 Normal Vector V1 V2 mx = Σ(yi -ynext(i)) (zi -znext(i)) V3 my = Σ(zi -znext(i))() (xi -xnext(i)) mz = Σ(xi -xnext(i)) (yi -ynext(i)) V5 V4 V2
V1 V1 V3 V2 V3
V5 V 4 V5
CSE 681 V4 Ray-Plane
Ax + by + cz + d = 0 n ·P = -d
P = s + t*v n· (s+t*v) = -d n·s + t*(n·v) = -d t = -(d + n·s)/(n·v)
CSE 681 Ray-Polyhedron
Polyhedron - volume bounded by flat faces Each face is defined by a ring of edges Each edge is shared by 2 and only 2 faces
The pol yhedron can be con ve x or conca ve
Faces can be convex or concave
CSE 681 Polyhedron Classification
Polyhedron
Convex Polyhedron Concave Polyhedron
Only One or more Only One or more Convex Concave Convex Concave Polygons Polygons Polygons Polygons
titriangul ltdated quads midixed titriangul ltdated quads midixed
CSE 681 Solid Modeling
Modeling of three-dimensional solids
Physically realizable objects
No infinitelyy, thin sheets, no lines
Interior of object should ‘hold water’ - Define a closed volume http://www.gvu.gatech.edu/~jarek/papers/SolidModelingWebster.pdf
CSE 681 Polygonal Solid Models
Vertices of a face have a consistent ordering (e.g. clockwise) when viewed from the outside side of the face
Each edge of a face is shared by one and only one other face Each edge appears oriented one way in one face and the other way in the other face
EULER’S FORMULA F – E + V = 2 F – E + V = 2 – 2P
CSE 681 Convex Polyhedron volume bounded by finite number of infinite planes
Computing intersections is similar to cube but using ray-plane intersection and arbitrary number of planes
n· P= -d Use n· v to determine P(t)=s+t*vP(t) = s + t*v Entering/exiting status
n· (s+t*)*v) = -d n·s + t*(n·v) = -d t = -(d + n·s)/(n·v)
CSE 681 Convex Polyhedron
Intersection going from outside to inside Intersection going from inside to outside
Record maximum entering intersection - enterMax Record minimum exiting intersection - exitMin If (enterMax < exitMin) polyhedron is intersected
CSE 681 Concave Polyhedron
Find closest face (if any) intersected by ray
Need ray -face ( ray -polygo n) inte rsect io n test
CSE 681 Ray-Convex Polygon E
Test to see if point is on n ‘inside’ side of each edge n·(VxE) > 0 V VxE Dot product of normal Cross product of ordered edge vector from edge source to point of intersection
CSE 681 Ray-Concave Polyhedron
1. Intersect ray with plane
2. Determine if intersection point is inside of 2D polygon
A) Convex polygon
B) Concave polygon
CSE 681 Ray-concave polygon Project plane and point of intersection to 2D plane 2D point-inside-a-polygon test N (lbdf(can also be used for convex pol ygons)
A PjProject to pl ane of f2 2 B smallest coordinates of normal vector
Form semi-infinite ray A’ and count ray-edge intersections B’
CSE 681 2D Point Inside a Convex Polygon p Semi-infinite ray test
(py < p1y )&&(py > p2y )|| (py > p1y )&&(py < p1y ) px < p1x + ( p2x − p1x )(py − p1y ) /(p2y − p1y )
Test to see if point is on ‘inside’ side of each edge p2 p Need to know order (CW or CCW) of 2D vertices p11
(px − p1x )(p2 y − p1y )− (py − p1y )(p2x − p1x )
CSE 681 2D point inside a polygon test
CSE 681 Special Cases Logically move ray up epsilon
N N
Y Y
CSE 681 2D point inside a polygon test
y
if (((y < y2 ) & &(y >= y1)) || ((y < y1) & &(y >= y2 )))
CSE 681 Transformeds o ed objecobjects s
e.g., Ellipse is transformed sphere
Objjpect space WldWorld space
Intersect rayjy with transformed object
Use inverse of object transformation to transform ray Intersect transformed rayyj with untransformed object
CSE 681 Transformed objects
r(t) = s + tv World space ray
s = [sx , sy , sz ,1]
v = [vx ,vy ,vz ,0]
M Object to world transform matrix
R(t)T = M −1sT + M −1vT Object space ray Intersect ray with object in object space
P T = MP T Transform intersection point and world object normal back to world space T −1 T T Nworld = (M ) Nobject
CSE 681 Ray-Cylinder
Q=sQ = s2 +t+ t2*v2
r P = s1 + t1*v1
At closest points
r (P(t1) − Q(t2 ))⋅v1 = 0
(P(t1) − Q(t2 ))⋅v2 = 0
CSE 681 Ray-Cylinder
(P(t1) − Q(t2 ))⋅v1 = 0
(P(t1) − Q(t2 ))⋅v2 = 0
(s1 + t1v1 − (s2 + t2v2 ))⋅v1 = 0
(s1 + t1v1 − (s2 + t2v2 ))⋅v2 = 0
CSE 681 Ray-Ellipsoid
GiGeometric constructi illon: all points p such that a b |p-a|||p + |p-b| = r d1 d2
Algebraic equation – axis aligned , origin centered
2 2 2 ⎛ x ⎞ ⎛ y ⎞ ⎛ z ⎞ ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ =1 ⎝ a ⎠ ⎝ b ⎠ ⎝ c ⎠
CSE 681 Ray-Quadidric
P(t) = (x, y, z) = s + tv = (sx + tvx , sy + tvy , sz + tvz )
Ax2 + By 2 + Cz 2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
http://en.wikipedia.org/wiki/Quadric
CSE 681