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Published byJonah Edwards Modified over 8 years ago
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Week 13 - Monday
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What did we talk about last time? Exam 2! Before that… Polygonal techniques ▪ Tessellation and triangulation Triangle strips, fans, and meshes Simplification ▪ Static ▪ Dynamic ▪ View-dependent
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We need to test for various intersections for lots of reasons What object is the mouse hovering over (or your gun pointing at)? (Called picking) Do these two objects collide? Does a ray hit a bounding box? As with everything in real time graphics, it needs to be efficient
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It's a challenging (and computationally expensive) task to see if a ray intersects with, say, a demon model Instead, models are often surrounded by bounding volumes that are easy to test intersection against If a ray or another model (or another model's bounding volume) intersects with a bounding volume, we can compute a more exact intersection If not, it doesn't matter
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Bounding Sphere Easy to do intersection tests with Poor fit for many rectangular objects More costly intersection tests Better fit for rectangular objects No computational savings at all Perfect fit for cow Bounding BoxBounding Cow
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A ray r(t) is defined by an origin point o and a direction vector d d is usually normalized Negative t values are behind the starting point of the ray and usually don't count If d is normalized, positive t values give the distance of the point from o It is also common to store l, the maximum distance along the ray we want to look
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An implicit surface is one described by a vector equation where any point on the surface has a value of 0 f(p) = f(p x, p y, p z ) = 0 Implicit sphere: f(p) = p x 2 + p y 2 + p z 2 - r 2 = 0 An explicit surface is one parameterized by two parameters Explicit sphere:
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An axis-aligned bounding box (AABB) (also known as a rectangular box) is a box whose faces have normals pointing the same way as the x, y, and z axes It's a non-rotated box in 3 space It can be defined with two points (lower corner and upper corner)
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An oriented bounding box (OBB) is an AABB that has been arbitrarily rotated It can be described by A center point b c Three normalized vectors b u, b v, and b w giving the side directions of the box And half-lengths (from center to wall) h u B, h v B, and h w B
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A k discrete oriented polytope (k-DOP) has k/2 normalized normals n i For each n i, there are two values d i min and d i max which defines a slab S i that is the volume between the two planes The k-DOP is the intersections of all the slabs
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For two arbitary, convex, disjoint polyhedra A and B, there exists a separating axis where the projections of the polyhedra are also disjoint Furthermore, there is an axis that is orthogonal to (making the separating plane parallel to) 1. A face of A or 2. A face of B or 3. An edge from each polyhedron (take the cross product) This definition of polyhedra is general enough to include triangles and line segments
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An AABB is the easiest Take the minimum and maximum points in each axis and, BOOM, you've got an AABB A k-DOP is not much harder Take the minimum and maximum values in each of the k/2 axes and use those to define the slabs You have to have axes in mind ahead of time
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Not as simple as you might think One approach: Make an AABB and use the center and diagonal of the corners to make your sphere Or: Make the AABB and do another pass through the vertices, taking the one furthest from the center as the radius There are other more complicated ideas
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The relative probability that a random point is inside an object is proportional to the object's volume However, the relative probability that a random ray intersects an object is proportional to its surface area
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Messiest yet! Find the convex hull of the object Find the centroid of the entire convex hull Compute a matrix using math from the book The eigenvectors of the matrix are the direction vectors of the box Use them to find the extreme points for each axis There are other approaches
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Perform computations early that could easily get a reject or accept If possible, reuse results from previous tests If you are using multiple tests, experiment with the order in which you apply them Postpone expensive calculations (trig, square roots, and divisions) until you absolutely need them Reduce the problem to a lower dimension whenever possible If one ray or object is being tested against many others, try to compute whatever you can for that one item a single time Expensive intersection tests should always come after some simpler BV test Use empirical timing data to see what really works best Write robust code, particularly with respect to floating point issues
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We can write the implicit sphere equation as f(p) = ||p – c|| – r = 0 p is any point on the surface c is the center r is the radius By substituting in r(t) for p, we can eventually get the equation t 2 + 2tb + c = 0, where b = d (o – c) and c = (o – c) (o – c) – r 2 If the discriminant is negative, the ray does not hit the sphere, otherwise, we can compute the location(s) where it does
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Looking at it geometrically, we can optimize the test Find the vector from the ray origin to the center of the sphere l = c – 0 Find the squared length l 2 = l l If l 2 < r 2, then o is in the sphere, intersect! If not, project l onto d: s = l d If s < 0, then the ray points away from the sphere, reject Otherwise, use the Pythagorean theorem to find the squared distance from the sphere center to the projection: m 2 = l 2 – s 2 If m 2 > r 2, the ray will miss, otherwise it hits
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There are a few basic primitives that SharpDX uses for intersection testing Plane Defined by a normal and a distance from the origin along that normal Or you can make one with three points Ray Defined by a starting point and a direction vector Vector3 Used (unsurprisingly) for points
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BoundingBox Axis aligned bounding box Defined by two points BoundingSphere Defined by a center point and a radius BoundingFrustum Defined by a matrix (view x projection)
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More intersection test methods Ray/box intersection Line segment/box overlap Ray/triangle intersection Plane/box intersection
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Keep working on Project 4 Keep reading Chapter 16
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