Spectrum of Computational Spaces Information Density Delaunay - - PowerPoint PPT Presentation

Local Global Nearest Neighbor Binary Space Partitioning Delaunay Triangulation / Voronoi Diagram

Information Density Spatial Range

A/B/D A/BE A/C A B C D E

Ideal Spaces Infinitesimal Infinite Homogeneous Heterogenous

Spectrum of Computational Spaces

Coordinates and Vectors

Addition Subtraction

• dot(A, B) = |A|*|B|*cos()

A B

|A|*|B|*cos() AxB = C, |C| = |A|*|B|*sin()

Dot Product Cross Product

A B C

answers the question: How much is a vector aligned with another? answers the questions: How orthogonal are two vectors? What plane do two vectors define?

Window Culling Camera Geometry

Rendering Pipeline

ModelView Projection Divide by W Viewport Transform

Object Coordinates Eye Coordinates Clip Coordinates Normalized Device Coordinates Window Coordinates

(x, y, z) (x, y, z, 1) (xe, ye, ze, 1) (xc, yc, zc, 1) (xn, yn, zn) (xw, yw, Zw)

glVertex glTranslate glRotate glScale glPerspective glOrtho glFrustum gluLookAt glViewport

x, y, z [-1, 1]

Normal Matrix Transpose(Inverse(MV))

glNormal

(x, y, z) (xe, ye, ze)

ModelView Matrix

X Y Z W=1 Mmodel Mview Mmodelview X Y Z W=1 Xeye Yeye Zeye Weye =

=

• 1

m22 m02 m12 T2 m11 m01 m21 T1 T0 m20 m10 m00 1 Z Y

X

Z Y

X

T0 T1 T2 m22 m11 m20 m10 m00 m02 m21 m01 m12

• +

Homogeneous Coordinates Canonical Affine Transformation

1 1 Tz 1 Ty Tx 1 glTranslate(Tx, Ty, Tz) (t0, t1, t2) (0, 0, 0)

Translation

1 R22 R02 R12 R11 R01 R21 R20 R10 R00 glRotate(angle, Rx, Ry, Rz)

Rotation angle

X

(Rx, Ry, Rz)

1 1

cos() sin()

• sin()

cos()

glRotate(, 0, 0, 1)

Rotation

X

• sin()

sin() cos() cos() X-axis Y-axis

(cos(), sin()) (-sin(), cos())

1

cos()

• sin()

1

• sin()

cos()

1

cos() sin() cos()

• sin()

1

Z-axis Y-axis X-axis roll Yaw Pitch

glRotate(Yaw, 0, 1, 0) glRotate(Pitch, 1, 0, 0) glRotate(Roll, 0, 0, 1)

Compound rotation of 3 angles:

1 Sz Sy Sx glScale(Sx, Sy, Sz)

Scale s0 s1 s2

1 Look(z) Look(x) Look(y)

Up(y) Up(x) Up(z) Left(z) Left(y) Left(x)

gluLookAt(eye, center, up) 1 m22 m02 m12 T2 m11 m01 m21 T1 T0 m20 m10 m00

Viewing Axes Current MV Left Up Look Eye Center Look = Center - Eye Left = Up X Look

• Note: viewing transforms can be thought of as opposing model

transforms, thus the transposed axes in the View Matrix

L M A B/B' C C' Eye A' L M

1D Projective Mapping

Desirable Properties: bijectivity

How to map line L to line M?

Ideal Point Ideal Point

w=0 w=1

(kx, ky, kz, kw) (x/w, y/w, z/w, 1)

Projective Plane (P2):

Set of all equivalence classes of ordered triples of non-zero vectors in E3 where equivalence is the mutual proportionality of two vectors Set of all lines passing through the origin of E3

E3 = 3D Euclidean Space P2 = Projective Plane P2 S2 = Unit sphere in E3 S2

(x, y, z, w) equivalent points

ideal lines

Fundamental Polygons

Möbius Strip Sphere

A A A A B B pole pole

Torus

A B A B

Projective Plane

A B A B