2.5. BASIC PRIMITIVE INTERSECTION
Details of common forms of primitive intersection test
Basic Primitive Intersection Tests
Basic primitive intersection testing concerns
determining if two primitives intersect.
Testing if two primitives intersect is often easier (and faster) than
determining how two primitives intersect (e.g. generating contact
information such as the distance of interpenetration, etc.).
Types of basic primitive test
Consult the recommended course text for details of the
following tests:
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•
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Separating-axis test
Sphere against plane
Box against plane
Cone against plane
Sphere against AABB
Sphere against OBB
Sphere against triangle
Sphere against polygon
AABB against polygon
Triangle against triangle
Two illustrative forms of closest point test are explored next.
Overview of the separating axis theory
Separating-axis test
The separating-axis test provides a core tool for implementing a range of
intersection tests.
The test is based on the separating hyperplane theorem, which states that
given two convex sets A and B, either the two sets are intersecting or there
exists a separating hyperplane P such that A is on one side of P and B is on
the other.
The theory is intuitive as two convex objects
cannot “curve” around each other. Thus, when
they are not intersecting there will be a gap
between them into which a plane can be inserted
to separate the two objects.
Separating-axis test
A separating axis is a line which is perpendicular to some separating
hyperplane.
It is called a separating axis because the orthogonal projections of A and B
onto the separating axis result in two non-overlapping intervals.
In principle, a test for a separating axis or hyperplane is equally valid,
however, in practice, it is less expensive to test for separation on an axis.
Separating-axis test: Symmetrical primitives
For symmetrical primitives with a defined centre point (e.g. AABBs, OBBs,
spheres, etc.), the centre point will always project onto the middle of the
projection interval along the tested axis.
An efficient separation test of
two symmetrical objects A and
B is to compute the halfwidth, or
radii, of their projection intervals
and compare the sum of them
against the distance between
their centre projections.
If the sum is less than the
distance between the centre
projections, the objects must be
disjoint.
Separating-axis test: Arbitrary primitives
Given two arbitrary convex hulls (i.e. polyhedra) the following forms of
intersection are possible: face-face, face-edge, and edge-edge.
For face-face and face-edge cases, the face normals of both objects should
be tested as potential separating axes. For edge-edge case, the cross
product of the two edges should be tested as the potential separating axis
(the points on the edges closest to each other form the feet of a
perpendicular between the two edges).
In summary, to determine intersection of two polyhedral objects, the
following axes should be tested for separation:
• Axes parallel to face normals of object A
• Axes parallel to face normals of object B
• Axes parallel to the vectors resulting from the cross
products of all edges in A with all edges in B
Aside: More fully: this list also
includes face-vertex, edgevertex and vertex-vertex,
however, the vertex
combinations can be considered
a special case of edge contact.
Separating-axis test: Arbitrary primitives
As soon as a separating axis is found, a test can exit immediately, i.e. it is
not until all identified axes have been tested with no separating axis that
the objects must be intersecting.
Aside: The separating-axis test
can also be used to derive
contact information, e.g. instead
of exiting early when an overlap
is detected, all axes are tested for
overlap. After all axes have been
tested, the axis with the least
(normalized) overlap can be used
as the contact normal, and the
overlap can be used to estimate
the penetration depth. With
some extra work, contact points
can also be computed with the
help of the separating axis.
Sphere and box plane-intersection tests
Sphere vs Plane Intersection
It is possible to test a sphere against
a plane in several ways, e.g. testing if
the sphere intersects the plane, or if
the sphere intersects the negative
halfspace of the plane.
bool TestSphereAndHalfspace(
Sphere sphere, Plane plane)
{
float separation =
Dot(sphere.Centre, plane.Normal)
- plane.Distance;
return separation <= sphere.Radius;
}
Box vs Plane Intersection
Given that the box vertices
represent the points furthest
away from the centre of the
box, they need only be
compared for distance against
the plane.
By projecting each vertex along
the plane’s normal and
accumulating the result, the
maximum projection radius is
obtained. Comparing this to the
distance of the OBB’s centre
from the plane determines if it
is in intersection.
e1
(P-C)● uo
u0
u1
●
Centre-plane
distance
C
e0
Projected
radius
n
●
e1
Box vs Plane Intersection
(P-C)● uo
u0
u1
●
C
e0
Centre-plane Projected
distance
radius
n
●
bool IsOBBIntersectingPlane(OBB box, Plane plane)
{
Accumulate the projections of each box half-width along the plane
float radius = normal to determine the total box radius along towards the plane.
box.e[0] * abs( dot( plane.n, box.u[0] ) ) +
box.e[1] * abs( dot( plane.n, box.u[1] ) ) +
box.e[2] * abs( dot( plane.n, box.u[2] ) );
Determine distance of box’s centre from the plane
float distance = dot( plane.n, box.c ) – plane.d;
Intersection if distance is within ±radius
return abs(distance) <= radius;
}
Intersection tests involving rays
Intersecting lines rays and segments
Consult the recommended course text for details of the
following tests:
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Intersecting segment against plane
Intersecting ray or segment against sphere
Intersecting ray or segment against box
Intersecting line against triangle
Intersecting line against quadrilateral
Intersecting ray or segment against triangle
Intersecting ray or segment against convex polyhedron
An illustrative form of ray test is explored next.
Ray or Segment vs. Sphere
Given a ray defined as R(t) = P + td, t ≥ 0, where P is the ray origin and d the
normalized direction vector (for a defined segment 0 ≤ t ≤ tmax). For a
sphere defined as (X-C)●(X-C)=r2, the point(s) of intersecti0n can be found
by substituting R(t) for X and solving for values of t, i.e.
(P+td-C)●(P+td-C)=r2
Rearranging the above provides a quadratic expression in terms of t, i.e.:
t2 + 2(m●d)t+(m●m)-r2 = 0, where m=P-C
Solving for t will enable the points of intersection to be determined, i.e. t =
-b±√(b2-c) where b=m●d and c=(m●m)-r2
Zero, one or two solutions will be found, negative solutions should be
ignored!
Ray or Segment vs. Sphere
bool IntersectRaySphere(
Point p, Vector d, Sphere s,
d
out float t, out Point q)
P●
{
Vector m = p - s.c; Determine b and c for use
float b = dot(m, d); within t = -b±√(b2-c)
float c = dot(m, m) - s.r * s.r;
t●
Q
Return false if the ray origin is outside of the sphere (i.e.
c>0) and pointing away from the sphere (b>0)
if (c > 0.0f && b > 0.0f) return false;
}
If the discriminant is
float discr = b*b - c;
negative, i.e. No real
if (discr < 0.0f) return false; solutions, then the ray
misses sphere
t = -b - sqrt(discr);
if (t < 0.0f) t = 0.0f; Determine the smallest (nonnegative) value of t which gives the
q = p + t * d;
nearest point of intersection. If
return true;
negative, then clamp to zero (ray
started within sphere)
r
C●
Directed reading on primitive intersection
Directed reading
• Read Chapter 5 of Real Time
Collision Detection:
o pp156-174 for primitive intersection
tests
o pp175-200 for line, ray and segment
intersection
o pp201-231 for dynamic and other
forms of intersection test
• Related papers can be found from:
http://realtimecollisiondetection.net/books/rtcd/references/
Summary
Today we
explored:
 Basic
primitive
intersection
tests
 Separating
axis theory
 Sample plane
and ray
intersection
tests