Constructive Solid Geometry (CSG) and other modelling techniques€¦ · COSC342 Constructive Solid Geometry (CSG) and other modelling techniques 21. Subdivision surfaces É A recursive

Constructive Solid Geometry (CSG)and other modelling techniques

COSC342

Lecture 2112 May 2015

There are many ways to model objects

É So far we have represented objects as sets of transformed, ray tracedprimitives.

É In this lecture we will explore some other techniques for modellingreal-world objects.

É Constructive Solid Geometry (CSG)

É Parametric Patches

É Subdivision Surfaces

É Implicit Surfaces

COSC342 Constructive Solid Geometry (CSG) and other modelling techniques 2

Constructive Solid Geometry (CSG)


Building a simple object using CSG

É A sequence of images from the (Otago) Mirage CSG engine built bystudent David Mason.


Direct ray tracing example


Our object and its CSG ‘tree’

É We show the two ‘subtract’ nodes in the CSG tree on the right.

É Yellow volumes are subtracted from our resulting shape.

−

−


Classifying the ray

É There are five intersections between the ray and CSG tree geometry:

É . . . these are labelled ‘a’ to ‘e’ in the figures below.

aedb c

−

−


Classification at a CSG subtract node

Right tree child

In Out Border

Lef

ttr

eech

ild In Out In Border

Out Out Out Out

Border Out Border “yuck”


Subtract nodes for our object

É Consider point ‘a’.

É It’s outside the hemisphere

É . . . and outside our object.

aedb c

a a

a

−

a

a

−


Right tree childIn Out Border

Lef

tt.

c. In Out In BorderOut Out Out Out

Border Out Border


É Now consider point ‘b’.

É It’s outside hemisphere

É . . . and outside our object

aedb c

b b

b

−

b

b

−



Lef

tt.


Border Out Border


É Next consider point ‘c’.

É It’s on hemisphere’s border

É . . . but still outside object

aedb c

c c

c

−

c

c

−



Lef

tt.


Border Out Border


É Finally, consider point ‘d’.

É It’s inside the hemisphere

É . . . on our object’s border.

aedb c

d d

d

−

d

d

−



Lef

tt.


Border Out Border

Classification at an add node

Right tree child

In Out Border

Lef

ttr

eech

ild In In In In

Out In Out Border

Border In Border


Pros and cons of CSG

É Ø Engineering objects can be created easily.

É Ø CSG is directly ray traceable.

É Ø CSG can be used to compute volume properties.

É ✗ It is difficult to make free-form shapes.


Parametric patches

É Polygons are flat.

É Most objects have curved faces.

É Can we fit a mathematical curve through particular points in space?

É Can we fit a surface to points in space?


Curve fitting

É We can define y = f (x), e.g. f (x) = c1(x1 − x)(x2 − x)(x3 − x) + c2É Alternatively, we can define a start and end point and two control

points that influence curvature.É This is still just four pieces of information.

X

Y

X

Y


But how do we do this one?

É We are likely to want to fitcurves to shapes where y isn’tjust a function of x .

É We can use a parameter (e.g. t).

É Define two functions y = f1(t)and x = f2(t).

É Aside: this figure is defined by:x(t) = 0.9 + 0.9t cos t andy(t) = 0.4 + t sin t

X

Y


Defining a circle using a parameter, θ

θ

X

θ

Y

X

Y


Patches

É Beyond generating a curve in space, we can generate a surface if weuse two parameters: e.g. s and t.

É (The uv coordinates used in texture mapping are a similar concept.)

s

t

É Coordinates of point p = (x , y , z) on the green patch are given by:x = fx(s, t) y = fy (s, t) z = fz(s, t)


Function for x in a bicubic patch

É If we use degree 3 polynomials in s and t to define the x coordinateof point p, we get the following form of equation:

fx(s, t) =

x3,3s3t3+x3,2s

3t2+x3,1s3t+x3,0s

3+x2,3s

2t3+x2,2s2t2+x2,1s

2t+x2,0s2+

x1,3st3+ x1,2st

2+ x1,1st+x1,0s+x0,3t

3+ x0,2t2+ x0,1t+x0,0

É As well as these sixteen constant xi ,j coefficients, we need coefficientsfor y and z , for 48 in total!

É Often these coefficients will be grouped into 16 points ui ,.j .


Pros and cons of parametric patches

É Ø We can make a huge variety of shapes.

É Ø Smooth surfaces are produced.

É Ø Spheres and cylinders can be represented well.É ✗ It is difficult to coordinate boundary conditions.

É e.g. patches meeting other patches.

É ✗ Parametric patches only define the surface—there’s no ‘inside’.


Subdivision surfaces

É A recursive approach to adding detail.

É For example, Catmull-Clark mesh subdivision:(steps are only sketched out here)

É Add a point at centroid of each face.É Add new ‘average’ edge points.É ‘Smooth out’ original vertices.

É In the limit, a cube becomes a sphere.


Implicit surfaces

É Instead of generating points p explicitly using parameters, e.g.

p =�

fx(s, t), fy (s, t), fz(s, t)�

É . . . implicitly define the conditions for a point p to be on a surface:f (p) = k

É For example, p2 = 1 defines a sphere.

É Usually f (p) will be a function of distance, such as the function below:

d

f (d)

maximum D


Example implicit surfaces

É Thermodynamics analogy: think of each red control point as a heater.

É The implicit surface is defined where the air temperature crosses acertain threshold.


Pros and cons of implicit surfaces

É Ø Intuitive overall construction concept.

É Ø Guaranteed inside/outside for objects—no holes.

É Ø Fairly easily ray traceable.

É Ø Can straightforwardly change topology.

É ✗ There is no obvious way to draw a surface.

É ✗ Good design seems to be by black art.


Examples of various modelling techniques

CSG in CAD Parametric surface teapot

Implicit surfaces “Swirling sweepers”


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Constructive Solid Geometry (CSG) and other modelling techniques€¦ · COSC342 Constructive Solid Geometry (CSG) and other modelling techniques 21. Subdivision surfaces É A recursive