cs 376 introduction to computer graphics 04 / 30 / 2007 last lecture :( instructor: michael eckmann

CS 376Introduction to Computer Graphics

04 / 30 / 2007Last Lecture :(

Instructor: Michael Eckmann

Michael Eckmann - Skidmore College - CS 376 - Spring 2007

Today’s Topics• Questions?• Radiosity discussion by Alex Danilevsky• Then you'll fill out evaluation forms for the course• Then I'll come back in the room and cover:

– Surfaces (bicubic polynomial surfaces)– Speedups for polynomial evaluation– quick comment on Shape Grammars

• probably no time for Fractal geometry


• Recall that we defined parametric cubic curves as Q(t) = T M G.• For surfaces we define a parametric cubic surface with two parameters,

hence Q(s,t)• for some particular value of s, say s

1, Q(s

1,t) is a parametric curve. Also,

for some particular value of t, say t1, Q(s,t

1) is a parametric curve.

• [ G1(t) ]

• Q(s,t) = S M G(t) = S M [ G2(t) ]

• [ G3(t) ]

• [ G4(t) ]

• where each Gi(t) is a parametric cubic . So, each G

i(t) = T M G

i

• The overall G(t) geometry matrix contains 16 geometry items (whereas for cubic curves, G contained 4 geometry items.)

Curve Surfaces


• Horner's rule for evaluating polynomials– consider the x-coordinate parametric cubic:

• x(t) = axt3 + b

xt2 + c

xt + d

x

• evaluated as is, operations used:– 6 multiplies and 3 adds

• if you use Horner's rule, x(t) can be rewritten as:• x(t) = [(a

xt + b

x)t + c

x]t + d

x

– 3 multiplies and 3 adds– for all 3 coordinates, then you have 9 multiplies and 9 adds

– we can do better

• Forward-difference calculations– generate successive values recursively by incrementing the previous

value– x

k+1 = x

k + Δx

k

– Δxk

is the forward difference

Speedups for polynomial evaluation


• Forward-difference calculations– generate successive values recursively by incrementing the previous

valuex

k+1 = x

k + Δx

k

– Δxk

is the forward difference– for parametric cubics if we divide our range of t into a set number of

fixed intervals of size δ thenx

k = x(t

k) and x

k+1 = x(t

k+1) where t

k+1 = t

k + δ k=0,1,...

xk = a

xtk3 + b

xtk2 + c

xtk + d

x

xk+1

= axtk+1

3 + bxtk+1

2 + cxtk+1

+ dx

= ax(t

k+δ)3 + b

x(t

k+δ)2 + c

x(t

k+δ) + d

x

Δxk = x

k+1- x

k = 3a

xδt

k2 + (3a

xδ2 + 2b

xδ)t

k + (a

xδ3 + b

xδ2 + c

xδ)

Notice that Δxk

is a quadratic polynomial in t



• we can do the same kind of thing but this is the second forward-difference

Δxk = Δx

k-1 + Δ

2x

k-1

– Δ2x

k-1 is the second forward difference

...Δ

2x

k-1 = 6a

xδt

k-1 + 6a

xδ3 + 2b

xδ2

Notice that Δ2x

k-1 is a linear polynomial in t

• we can do the same kind of thing but this is the third forward-difference

Δ2x

k-1 = Δ

2x

k-2 + Δ

3x

k-2

– Δ3x

k-2 is the third forward difference

...Δ

3x

k-2 = 6a

xδ3 which is a constant in t



x(t) = axt3 + b

xt2 + c

xt + d

x

Δxk = 3a

xδt

k2 + (3a

xδ2 + 2b

xδ)t

k + (a

xδ3 + b

xδ2 + c

xδ)

Δ2x

k-1 = 6a

xδt

k-1 + 6a

xδ3 + 2b

xδ2

Δ3x

k-2 = 6a

xδ3

• We need initial values for x and the first and second forward differenceswhen k = 0, t = 0 and we get the followingx

0 = d

x

Δx0 = a

xδ3 + b

xδ2 + c

xδ

Δ2x

0 = 6a

xδ3 + 2b

xδ2

and compute the first few x's and then

• to compute successive values of x it only takes 3 adds at each step:x

k+1 = x

k + Δx

k Δx

k = Δx

k-1 + Δ

2x

k-1

Δ2x

k-1 = Δ

2x

k-2 + Δ

3x

k-2



• To see the forward differences algorithm, see the handout with the two

algorithms (the first is brute force --- not even using Horner's rule).

• The brute force uses 11 multiplies and 10 additions.

• Horner's rule algorithm requires 9 multiplies and 10 additions.

• The forward differences algorithm only takes 9 additions (and 0 multiplies)

to calculate the next point coordinates (after some initialization that takes

some multiplies and adds.)



• See section 8-24 in our text. It's short. I suggest you read it and attend

Tim Gildea's presentation of his thesis "LSystem Viewer: Interactive

Visualization Software for Parametric DOL-Systems" which is

at 8pm, TONIGHT, Monday April 30th in Harder room 202.

• Two other students will present their work before Tim.

• 6:30 p.m. Kellen Affleck

• 7:15 p.m. Max Levine

• 8:00 p.m. Tim Gildea

Shape Grammars


• Up until now, we have only discussed objects (Euclidean shapes) being

described by equations. These methods are good for man-made objects

that are smooth and have regular shapes.

• Natural objects like mountains, clouds, terrain, trees, etc. have surface

shapes that aren't well described by equations. Instead these are better

described by “procedures”. Fractal methods are a way to make a

procedural description of a surface/object.

• Fractal objects have 2 important characteristics– infinite detail at every point– self-similarity between the parts to the overall features (see next slide)

• Fractal objects are specified by a procedure that does an operation

repeatedly.

Fractal Geometry


• Zooming in enough on an equation defined surface (Euclidean shape) will

reveal the limit to the level of detail in the shape.

• Zooming in on a Fractal object will reveal the “limitless” detail in the

shape.

• (I'll try to draw figure 8-67 in text pg 480.)

• Fractals are defined infinitely, but of course in graphics we will perform a

finite number of steps and hence our objects will have finite dimensions.

Fractal Geometry


• Self-similarity– entire object contains scaled down versions of itself as its subparts– each subpart is created by multiplying a scale factor by the whole

object– each subpart can use the same scale factor or different ones– random variations can be employed -> statistically self-similar

• used for trees, leaves, etc.

Fractal Geometry


• Fractal Dimension D– a description of the amount of variation in the structure of a fractal

object.• a line is 1 dimensional, a square is 2 dimensional• if we subdivide a line into N equal parts, each part is the original

scaled down by a factor of N = N1/1 = N1/d

• if we subdivide a square into N equal parts, each part is the original scaled down by a factor of sqrt(N) = N1/2 = N1/d

• if we subdivide the Koch snowflake into 4 equal parts, each part is the original scaled down by a factor of 3, hence 3 = 41/d = 41/1.26186

• The dimension of the Koch snowflake fractal = d = log(4)/log(3) which is approx. 1.2618595

• See the slide showing the Koch snowflake

Fractal Geometry


• For deterministic, self-similar fractals we have

• Fractal initiator– start shape

• Fractal generator– the pattern that subparts of the initiator are replaced by

• The generation of the fractal is deterministic because we always replace a

subpart of the initiator with the generator and we do this at each step

exactly the same way every time.

• Example on the board (figure 8-70).

Fractal Geometry


• For statistically, self-similar fractals we could have– a Fractal initiator and – randomly choose a generator from a set of generators to be the pattern

to replace a subpart of the initiator at each step

• This is non-deterministic (won't get the same exact shape every time) due

to the random choice of generator.

• Instead of random generators, – random scalings could be done or – something else that is random to make the fractal non-deterministic.

• see figure 8-80 for an example.

Fractal Geometry

cs 376 introduction to computer graphics 04 / 30 / 2007 last lecture :( instructor: michael eckmann

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