Game Math Case Studies

Eric Lengyel, PhDTerathon Software

Content of this Talk

Real-world problems from game dev

Small problems, that is, and easy to state

Actual solutions used in shipping games

Strategies for finding elegant solutions

Occlusion boxes

Plain boxes put in world as occluders

Extrude away from camera to form occluded

rendered

How to do this most efficiently?

Camera

Occluder

Occlusion boxes

Could classify box faces as front/backand find silhouette edges

Similar to stencil shadow technique

A better solution accounts for smallsolution space

Occlusion boxes

There are exactly 26 possible silhouettes

Three possible states for camera position on three different axes

position < box min

position > box max

box max

Inside box excluded

maxx xminx x

min maxx x x

min maxy y y

miny y

maxy y

condition code

x > xmax 0x01

x < xmin 0x02

y > ymax 0x04

y < ymin 0x08

z > zmax 0x10

z < zmin 0x20

Finite classifications

Marching Cubes, fixed polarity(256 cases, 18 classes)

Transvoxel Algorithm(512 cases , 73 classes)

http://www.terathon.com/voxels/

Occlusion boxes

Calculate camera position state and usetable to get silhouette

Always a closed convex polygon withexactly 4 or 6 vertices and edges

Occlusion boxes

0

1

23

45

6 7

x

yz

// Upper 3 bits = vertex count, lower 5 bits = polygon index

const unsigned_int8 occlusionPolygonIndex [43] =

{

0x00 , 0x80, 0x81, 0x00, 0x82, 0xC9, 0xC8, 0x00, 0x83, 0xC7, 0xC6, 0x00, 0x00, 0x00, 0x00, 0x00 ,

0x84 , 0xCF, 0xCE, 0x00, 0xD1, 0xD9, 0xD8, 0x00, 0xD0, 0xD7, 0xD6, 0x00, 0x00, 0x00, 0x00, 0x00,

0x85 , 0xCB, 0xCA, 0x00, 0xCD, 0xD5, 0xD4, 0x00, 0xCC, 0xD3, 0xD2

};

// All 26 polygons with vertex indexes from diagram on left

const unsigned_int8 occlusionVertexIndex [26][6] =

{

{ 1, 3, 7, 5},

{ 2, 0, 4, 6},

{ 3, 2, 6, 7},

{ 0, 1, 5, 4},

{ 4, 5, 7, 6},

{ 1, 0, 2, 3},

{ 2, 0, 1, 5, 4, 6},

{ 0, 1, 3, 7, 5, 4},

{ 3, 2, 0, 4, 6, 7},

{ 1, 3, 2, 6, 7, 5},

{ 1, 0, 4, 6, 2, 3},

{ 5, 1, 0, 2, 3, 7},

{ 4, 0, 2, 3, 1, 5},

{ 0, 2, 6, 7, 3, 1},

{ 0, 4, 5, 7, 6, 2},

{ 4, 5, 1, 3, 7, 6},

{ 1, 5, 7, 6, 4, 0},

{ 5, 7, 3, 2, 6, 4},

{ 3, 1, 5, 4, 6, 2},

{ 2, 3, 7, 5, 4, 0},

{1 , 0, 4, 6, 7, 3},

{ 0, 2, 6, 7, 5, 1},

{ 7, 6, 2, 0, 1, 5},

{ 6, 4, 0, 1, 3, 7},

{ 5, 7, 3, 2, 0, 4},

{ 4, 5, 1, 3, 2, 6}

};

Occlusion boxes

Any silhouette edge that is off screen can be eliminated to make occlusion region larger

Gives occluder infinite extent in that direction

Allows more objects to be occluded because they must be completely inside extruded silhouette to be hidden

Occlusion boxes

Silhouette edge is culled if both vertices on negative side of some frustum plane

And extruded plane normal and frustum plane normal have positive dot product

Camera

Occluder

Occlusion boxes

Strategy:

Look for ways to classify solutions

Oblique near plane trick

Sometimes need a clipping planefor a flat surface in scene

For example, water or mirror

Prevent submerged objects fromappearing in reflection

Ordinary frustum Oblique near plane

Oblique near plane trick

Hardware clipping plane?

May not even be supported

Requires shader modification

Could be slower

Oblique near plane trick

Extra clipping plane almostalways redundant withnear plane

Oblique near plane trick

Possible to modify projection matrix

Move near plane to arbitrary location

No extra clipping plane, no redundancy

Oblique near plane trick

In normalized device coordinates (NDC),near plane has coordinates (0,0,1,1)

Oblique near plane trick

Planes (row antivectors ) are transformed from NDC to camera space by right multiplication by the projection matrix

So the plane (0, 0, 1, 1) becomesM 3 + M 4, where M i is the i- th row of the projection matrix

Oblique near plane trick

M 4 thatperspective correction still works right

Let C = ( Cx, Cy, Cz, Cw) be the camera -space plane that we want to clip against

Assume Cw < 0, camera on negative side

We must have C = M 3

Oblique near plane trick

M 3 = C M 4 = ( Cx, Cy, Cz + 1, Cw)

This matrix maps points on the plane Cto the plane z in NDC

0 0 0

0 0 0

1

0 0 1 0

x y z w

e

e a

C C C CM

Oblique near plane trick

But what happens to the far plane?

F = M 4 M 3 = 2 M 4 C

Near plane and (negative) far plane differonly in the z coordinate

Thus, they must coincide where theyintersect the z = 0 plane

Oblique near plane trick

Oblique near plane trick

Far plane is a complete mess

Depths in NDC no longer represent distance from camera plane, but correspond to some skewed direction between near and far planes

We can minimize the effect,bad

Oblique near plane trick

We still have a free parameter:the clipping plane C can be scaled

Scaling C has the effect of changing the orientation of the far plane F

We want to make the new view frustum assmall as possible while still including the conventional view frustum

Oblique near plane trick

Let F = 2 M 4 aC

Choose the point Q which lies furthest opposite the near plane in NDC :

Solve for a such that Q lies in plane F:

1 sgn ,sgn ,1,1x yC CQ M

4a

M Q

C Q

Oblique near plane trick

planebecomes optimal

Oblique near plane trick

Works for any perspective projection matrix

Even with infinite far depth

More analysis available Oblique Depth Projection and View Frustum Clippingof Game Development, Vol. 1, No. 2.

http://www.terathon.com/lengyel/Lengyel-Oblique.pdf

Oblique near plane trick

Strategy:

Get the big picture

Fog bank occlusion

Consider fog bank bounded by plane

Linear density gradient with increasing depth

Perpendicular to plane

Zero density at plane