3d game engine design 1 3d game engine design ch. 2.5. 3d map lab

3D Game Engine Design 1

3D Game Engine Design3D Game Engine DesignCh. 2.5.Ch. 2.5.

3D MAP LAB


2.5. Standard 3D Objects2.5. Standard 3D Objects Useful as bounding regions for

Rapid culling in the rendering process Rapid determining of the collision

detection of two objects

Primitives Spheres Oriented Boxes Capsules, Lozenges Ellipsoids and etc.


Center Point and Radius with points

Sphere Containing Axis-Aligned Box Compute the minimum-volume axis-

aligned bounding box Select the smallest enclosing sphere of

the box with centered at the box center Rapid, but not as good a fit.

SpheresSpheresn

iiV

0}{

22

rcx


Spheres Spheres (cont’d)

Sphere Centered at Average of Points Sphere Center : Average of Points Sphere Radius : Smallest value from cent

er to points enclosing all points 2x Iteration than Axis-Aligned box method

Minimum-Volume Sphere (Welzl 91) Uses a randomized linear algorithm Maintain a set of supporting points( lying

on the sphere) while processing the input point set at a time.


Oriented BoxesOriented Boxes

Oriented Box Provide a better fit than spheres Defined by

A center 3 Orthonormal Axes for 3 extents of Axes

, points inside or on the box Orthonormal matrix where

C

i U

0 i e2, 1, 0 i

YR C X

X

) , , (2 1 0y y y Y

] , , [2 1 0U U U R


Oriented Boxes Oriented Boxes (cont’d)

Axis-Aligned Box Find the extreme points and

Fitting with a Gaussian Distribution Gaussian Distribution

: mean of the distribution : the covariance matrix of the distributi

on Fitting points with anisotropic gaussian di

stribution

min P

max P

)) ( ) exp((1

C X M C X AT

C

M



Fitting with a Gaussian Distribution

Axes of the box : unit-length eigenvectors of covariance matrix M

Extents :

n

jj V

nC

0

1 T

j

n

jjC V C V

nM) () (

10

) ( maxC V U ej i j i



Minimum-Volume Box Best-fitting box : the minimum volume con

taining the points Iterative scheme for a minimization

For any coordinate axes , points are projected onto the axes

The values :

i Ai As V

0

) (0 V V Aj i ij

) ( minij j i ) ( maxij j i 2

) (i ii



Iterative Minimization for M-V Box Center of the minimum-volume box

Extents of the minimum-volume box

Selection of coordinate axes : unit length vector and an

angle The volume of Box : Find that minimizes the volume

2

00

jj jA V C

2/ ) (i i i e

i U

) , ( ) , (0 0 U U

i A

) , ( U

i U



Fitting Triangles with Gaussian Distribution Data points : vertices of a triangle mesh Fit of an oriented box to the convex hull of the v

ertices triangles , i-th triangle Triangle and its interior

Mean point of the convex hull

l i i i oV V V., 2 ., 1 .,, ,

1 0 , 1 0 for ), ( ) ( ) , (., ., 2 ., ., 1 ., t s V Vt V Vs V t s Xi o i i o i i o i

2

0,

1

0 3

1j

i j

l

ijV

lC



Fitting Triangles with Gaussian Distribution Covariance matrix of the convex hull

Axes of Oriented Box :

unit length eigenvectors of M Extents of Oriented Box :

2

0

2

0, ,

1

0

) )( (12

1jk

T

i k i j

l

ijC V C V

lM

i A

) ( maxC X A ej i j i


CapsulesCapsules Capsule

The set of all points that are distant from a line segment with end point and direction

Shape : a cylinder that has 2 hemispherical caps at the end points

Least square fitting Line fitting : , where average of

the data points and unit-length direction vector

to be the maximum distance from the data points to Line

0 rPD

WtA

i AW

0 r


Capsules Capsules (cont’d)

Least Square Fitting Select unit axes vectors and set

Data points Find the domain of for the 2

hemisphere at end

Find the largest so that all points lie above the hemisphere

V &

UYR A X

] , [1 0 t

0 2 2

0

2 2) (r w v u

}) ( min{2 2 2

0i i iv u r w

0


Capsules Capsules (cont’d)

Least Square Fitting Find the smallest so that all points lie

below the hemisphere

End points :

Minimum of minimum-Area Projected Circles Projection of points onto the Plane Compute the minimum-area circle of

projected points and find and

1

}) ( max{2 2 2

1i i iv u r w

W A Pj j

0 X W

0 r ] , [1 0 t


LozengesLozenges

Lozenges Set of all points that are distant

from a rectangle with origin and edge directions

Shape : an oriented rectangle with Attached 4 half-cylinder sides 4 quarter-spherical corners

0 rP1 0 and E E


LozengesLozenges

Fitting with a Gaussian Distribution Compute the mean of the points and t

he covariance matrix M Unit-length eigenvectors of the M : Data points : Data points are bounded by 2 planes

Lozenge radius : A

W V, U

and ,

i i i i i i iWw Vv Uu A X

min ) (w A X W

max ) (w A X W

2/ ) (min maxw w r


Lozenges Lozenges (cont’d)

Fitting with a Gaussian Distribution Lozenge edge : end of the hem

icylinder Find the Lozenge edge

Analogous to the fitting by 3D capsule 2D capsule containing

The largest so that all points lie above the hemicircle

1 0 and E E

) , (i iw v

2 2

0

2 2) (r v w

0

} { min2 2

0i

i iw r v


Lozenges Lozenges (cont’d)

Fitting with a Gaussian Distribution Find the Lozenge edge

The smallest so that all points lie below the hemicircle

The Lozenge edge1

} { max2 2

1i

i iw r v

V E

) (0 1 1

U E

) (0 1 0


CylindersCylinders

Cylinder A distance from a line

Least Squares Line Contains Axis Fit the points by a line using the least-

square algorithm , where average of the data

points and unit-length direction vector Select unit axes vectors and set

Data points

0 r Dt P

R t

WtA

WtA

i A W

V, U YR A X


Cylinders Cylinders (cont’d)

Least Squares Line Contains Axis Cylinder radius : Cylinder height : Modified Axis

Least Squares Line Moved to a Minimum-Area Center Minimum-area circle containing is

computed and has center

} max{2 2

i iv u r min maxw w h

Www

AA

2maxmin

) , (i iv u) , (v u


EllipsoidsEllipsoids

Ellipsoid Center Orthonormal axis directions The ellipsoid with center and axes

C

2 0 for i Ui

1 ) ( ) ( C X DR R C XT T

} / 1, / 1, / 1{2

2

2

1

2

0d d d diag D

i UC

i U


Ellipsoids Ellipsoids (cont’d)

Axis-Aligned Ellipsoid Firstly generate the axis-aligned box cont

aining the points be the vectors of the min/m

ax box components Compute the ratios of semi-axis lengths

Semi-Axis Lengths

max min and P P

2/ ) (max minP P C

) , , ( 2/ ) (2 1 0 min max P P

} / 1, / 1, / 1{

)} ( ) {( max /

2 1 0 diag D

C V D C V D DR R Mi

T

i i

T

0



Fitting Points with a Gaussian Distribution Similar to the case of an oriented box Center of the ellipsoid : the mean of points Axes : eigenvectors of the covariance matrix Semi-Axis Lengths : eignevalues

} / 1, / 1, / 1{

)} ( ) {( max /

2 1 0 diag D

C V DR R C V DR R Mi

T T

i i

T



Minimum-Volume Ellipsoid Random Linear Techniques (Welzl 91) Constrained numerical minimization

Difficult to implement Not possible for real-time apps.

3d game engine design 1 3d game engine design ch. 2.5. 3d map lab

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