shadows in computer graphics - colorado college

Shadows in Computer Graphics

Steven Janke

November 2014

Steven Janke (Seminar) Shadows in Computer Graphics November 2014 1 / 49

Shadows (from Doom)


Simple Shadows


Shadows give position information


Shadow Geometry


Camera Model


View Frustum

Camera

Near

Far

View Plane

H0,0,0L

-z


Ideal City (circa 1485)


Perspective Projection

y

zE W

C C*

D

E = (0, 0, e),C = (x , y , z),C ∗ = (xs , ys)

EW = e,CD = y ,ED = e − z

4EWC ∗ ∼ 4EDC ys =y · ee − z

=y

1− ze


Albrecht Durer (Man drawing a Lute - 1525)


Alberti Perspective Diagram

1. Draw lines from front tile corners to center point C.

2. Select point R on horizontal line so CR is distance to painting.

3. Connect R with front tile corners.

4. Draw horizontal lines through intersections of lines in 3 and verticalline through C.

5. Diagonals through tiles are projected into diagonals.


Pedro Berreuguete - Anunciation (circa 1500)


Masaccio - Trinity (1426)


Calculating Cube Perspective

ys =y

1− ze

xs =x

1− ze

Example (Cube Vertices)

Eye Coordinates: (0,0,4)

World Coordinates:

(1, 1, 1), (1,−1, 1), (−1,−1, 1), (−1, 1, 1)

(1, 1,−1), (1,−1,−1), (−1,−1,−1), (−1, 1,−1)

Screen Coordinates:

(1.33, 1.33), (1.33,−1.33), (−1.33,−1.33), (−1.33, 1.33)

(0.75, 0.75), (0.75,−0.75), (−0.75,−0.75), (−0.75, 0.75)


Cubes in Perspective


Projections

Summary

Perspective projections and Shadow projections are both projections froma point onto a plane.

Next steps:

For shadows, we generalize projection to arbitrary plane.

Describe calculations compactly and efficiently.


Vectors

~v = (x1, y1, z1) and ~w = (x2, y2, z2) are displacements.

Algebra: ~v + ~w = (x1 + x2, y1 + y2, z1 + z2) and a~v = (ax1, ay1, az1).

Dot Product: ~v · ~w = |~v ||~w |cos(θ) = x1x2 + y1y2 + z1z2.(If ~v and ~w are perpendicular, then ~v · ~w = 0.)

Cross Product:

~v × ~w =

∣∣∣∣∣∣~i ~j ~kx1 y1 z1x2 y2 z2

∣∣∣∣∣∣ =

∣∣∣∣y1 z1y2 z2

∣∣∣∣~i − ∣∣∣∣x1 z1x2 z2

∣∣∣∣~j +

∣∣∣∣x1 y1x2 y2

∣∣∣∣~k


Vector Products

v

w

v -w

Θ

Hx1,y1L

Hx2,y2LH0,0L

Dot Product

A

B

A´B

Cross Product


Transformations

T~v =

a11 a12 a13a21 a22 a23a31 a32 a33

xyz

=

a11x + a12y + a13za21x + a22y + a23za31x + a32y + a33z

Rotation around z-axis uses this matrix:

Rz =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

Matrices give linear transformations: T (~v + ~w) = T~v + T ~w andT (a~v) = aT~v

Cannot represent translations or projections.


Homogeneous Coordinates

The point P0 = (−1, 5) is on the 2D line 3x + 2y = 7.

The vector equation of the line:

(3, 2) · (P − P0) = (3, 2) · (x + 1, y − 5) = 0

Let P = (x , y) = ( xhwh, yhwh

) be any point on the line 3x + 2y = 7.

=⇒ 3xh + 2yh − 7wh = (3, 2,−7) · (xh, yh,wh) = 0

(xh, yh,wh) are homogeneous coordinates for the point P. Since wh isarbitrary, there are infinitely many sets of homogeneous coordinatesrepresenting P. For example,

P0 = (−1, 5, 1) = (−2, 10, 2) = (−0.5, 2.5, 0.5)

Two-dimensional Homogeneous Line equation: ~n · P = 0


Homogeneous Coordinates for Lines

Example (2D Line coordinates)

P1 = (3, 2, 1) and P2 = (5, 7, 3) determine a two-dimensional line.

~n · (3, 2, 1) = 0 and ~n · (5, 7, 3) = 0.

~n = (3, 2, 1)× (5, 7, 3) = (−1,−4, 11)

The homogeneous coordinates (−1,−4, 11) represent the line.

Both points and lines in two dimensions can be represented byhomogeneous coordinates (x , y ,w).


Calculating with Homogeneous Coordinates

Example (2D Intersection Point)

Consider two lines: (2, 2,−1) and (6,−5, 2)(They represent the lines 2x + 2y − 1 = 0 and 6x − 5y + 2 = 0)

P is the point of intersection.

(2, 2,−1) · P = 0 and (6,−5, 2) · P = 0

P must be a vector perpendicular to the two homogeneous linevectors.

P = (2, 2,−1)× (6,−5, 2) = (−1,−10,−22) is the cross product.

P = (−1,−10,−22) represents the Cartesian point P = ( 122 ,

1022)


Points at Infinity

Lines 2x + 4y − 8 = 0 and 2x + 4y − 10 = 0 are parallel.

The homogeneous point (4,−2, 0) is on both lines.

Points of the form (xh, yh, 0) are points at infinity.

Notice that (4,−2, 0) and (5, 3, 0) are distinct points at infinity.

Points in 2D

Points at Infinity


Homogeneous Coordinates in Three Dimensions

Homogeneous coordinates for three dimensional points add a fourthcoordinate:

Cartesian (x , y , z) =⇒ Homogeneous (x , y , z , 1) or (tx , ty , tz , t)

Since planes are determined by a normal and a point, (tx , ty , tz , t)also represents a plane.

Homogeneous plane equation: ~n · P = 0

Lines have homogeneous coordinates called Plucker coordinates.


Perspective Matrix

Now we can express the perspective transformation as a matrixmultiplication:

T (P) = MP =

1 0 0 00 1 0 00 0 0 00 0 −1

e 1

xyz1

=

xy0

1− ze

In the space of homogeneous coordinates (Projective Space), theperspective transformation is a linear function.


Perspective Drawing


Two Point Perspective


Projective Geometry

Girard Desargues (1591 - 1661) was the founding father.

Parallel lines intersect in a point at infinity: (xh, yh, 0)

Points at infinity fall on a line.

Duality: In 2D, for any theorem about points there is a theoremabout lines.

No concept of length or angle.

Projective transformations are linear transformations.

In 2D, there is a projective transformation that sends a given fourpoints to another specified four points.


Desargues Theorem


Additional Vector Algebra

Tensor Product

~v ⊗ ~w = ~v ~wT =

vxvyvz

[wx wy wz

]=

vxwx vxwy vxwz

vywx vywy vywz

vzwx vzwy vzwz

Vector Triple Product

~A× (~B × ~C ) = (~A · ~C )~B − (~A · ~B)~C

Dot to Tensor(~A · ~C )~B = (~B ⊗ ~A)~C


2D Projection

PP’=T HPL

E

L

Line through E and P is E × P.

T (P) = P ′ = ~L× (E × P)

= (~L · P)E − (~L · E )P

= ((E ⊗ ~L)− (~L · E )I )P

= MP


2D Projection

Example

Project P = (3, 1) onto the line 6x + y − 5 = 0 from the point (8, 2).

~L = (6, 1,−5) E = (8, 2, 1) P = (3, 1, 1)

M = (E ⊗ ~L)− (~L · E )I

=

48 8 −4012 2 −106 1 −5

−45 0 0

0 45 00 0 45

=

3 8 −4012 −43 −106 1 −50

T (P) = MP =

3 8 −4012 −43 −106 1 −50

311

=

−23−17−31

Cartesian coordinates for P are (23/31, 17/31).


3D Projection

n

P

P’

E

P ′ = αP + βE .

~n · (αP + βE ) = 0 =⇒ α =−β(~n · E )

(~n · P)

P ′ =−β(~n · E )

~n · PP + βE

P ′ = T (P) = (~n · P)E − (~n · E )P = (E ⊗ ~n)P − (~n · E )P

=⇒ M = (E ⊗ ~n)− (~n · E )I


3D Projection

Example

Eye point: E = (7, 2, 6)Vertex: P = (4, 5, 0) Plane: 2x − y + 2z = −4

~n = (2,−1, 2, 4) E = (7, 2, 6, 1) P = (4, 5, 0, 1)

M = (E ⊗ ~n)− (~n · E )I =

14 −7 14 284 −2 4 8

12 −6 12 242 −1 2 4

−

28 0 0 00 28 0 00 0 28 00 0 0 28

=

−14 −7 14 28

4 −30 4 812 −6 −16 242 −1 2 −24

=⇒ P ′ = (−63,−126, 42,−21)

This gives Cartesian coordinates (3, 6,−2)Steven Janke (Seminar) Shadows in Computer Graphics November 2014 34 / 49

Compare to earlier 3D matrix

M = (E ⊗ ~n)− (~n · E )I

Perspective projection: Viewing plane is the xy plane with homogeneousrepresentation ~n = (0, 0, 1, 0). The eye point is E = (0, 0, e, 1) andP = (x , y , z , 1).

M =

−e 0 0 00 −e 0 00 0 0 00 0 1 −e

= −e

1 0 0 00 1 0 00 0 0 00 0 −1

e 1


Shadow Geometry


Points in Shadow

Shadow: Fill in the polygon determined by projected vertices.

For convex objects, shadows are convex.

Vertices that are not incident on both visible and hidden faces areinside the shadow.


Multiple Shadows


Ray Tracing

Camera

View Window

To Light Source

Reflected Ray

v


Shadow Ray

Light

P

Shadow


Soft Shadows


Penumbra & Umbra

Light

Object

Umbra

Penumbra Penumbra


More Soft Shadows


Multiple Lights


Shadow Volume


More Multiple Lights


Further Refinements

Shadow Maps

Curves (NURBS)

Complex Lighting Models (Radiosity)


Complex Shadows


Reference (2015


shadows in computer graphics - colorado college

Documents