Constructive Methods in ModellingConstructive Methods in Modelling

Lecture 7 (Modelling)

Surfaces of RevolutionSurfaces of Revolution

Creates circular symmetric objects.

Create a 3D surface by revolving a 2D profile curve around an axis of rotation in space.

Closed profile curves generate closed surfaces.

Examples:1. Circular cylinder: The profile

is a line segment parallel but not coincident with the axis of rotation. The closed version requires a rectangle.

2. Truncated cone: The line segment profile is slanted with respect to the axis.

Examples: Surfaces of RevolutionExamples: Surfaces of Revolution

3. Torus: The profile is a circle inset in a plane aligned with the axis.

4. Complex circularly symmetric shapes: employ a Bézier or B-spline profile curve.

Generalized CylindersGeneralized Cylinders

Extrusion: sweep a 2D shape along a (non-circular) path.

Some objects can be generated by extrusion or revolution. e.g. a cylinder (an extruded circle or a revolved line).

Generalized cylinders extend the concept of extrusions and surfaces of revolution to the extreme.

Total control over all sweep parameters.

But can produce degeneracies, e.g. self-intersection.

Generalized Cylinder ParametersGeneralized Cylinder Parameters

Can vary:

1. Cross Section – 2D shape does not have to be a circle and can change shape as it is swept.

2. Sweep Path – path does not have to be a straight line or revolution, can be any space curve.

3. Twist – the cross section can be rotated as it moves along the path

4. Scale – the size of the cross section can change along the path

5. Normal vector direction – conventionally the vector normal to the cross section points along the path, but even this can be varied

6. And any other parameters.

Spatial DeformationSpatial Deformation

Principle: Indirectly deform an object by warping the

surrounding space.

Jelly metaphor: A shape is set within a block of jelly.

Flexing the jelly results in a corres- ponding distortion of the shape.

Mechanism: Object vertices are embedded in a

parametric hyperpatch, which is a 3D generalization of B-spline curves.

Free Form DeformationFree Form Deformation

Hyperpatch Equation: Vertices (q) are expressed as a sum of control points (p)

weighted by B-spline basis functions (b).

FFD Algorithm:[1] Establish a (u,v,w) hyperpatch parameterization for

all object vertices.

[2] Displace control points.

[3] Apply the hyperpatch equation.

k,j,ik,j,ikji

pw,v,ubw,v,uq wbvbubw,v,ub kjik,j,i

IllustratIllustration of FFDion of FFD

Pre-Deformation Post-Deformation

EvaluatEvaluation of FFDion of FFD

Advantages: Fast Smooth, sculpted results Variable scope

Disadvantages: Lack of precise control Screen clutter Counter-intuitive

Later Developments:1. Direct Manipulation with points or curves

2. Different deformation boundaries.

Constructive Solid GeometryConstructive Solid Geometry

Constructive Solid Geometry (CSG) consists of regularized boolean set operations on closed 3D objects.

Boolean Set Operations

Union BA *

Intersection BA *

Difference BA *

Sphere (A) and Parallelepiped (B)

CSG: RobustnessCSG: Robustness

Regularized (represented by *): closed input objects produce a closed output object.

CSG arguments may be analytic (RT) or polygon mesh (PSC) objects (or both).

CSG is prone to special cases which cause robustness problems. If points, edges, or faces of two polygon mesh arguments coincide

an incorrect object may result. For example, in modelling a gaming die, if the top of the cylindrical

pips are flush with the cube surface then the difference may not be visible.

CSG is a huge area of research. Entire books have been written on this topic.

CSG TreeCSG Tree

CSG is evaluated as a pre-process in PSC and at run-time for RT.

The CSG tree is an effective structure for evaluating ray-object intersections.

Leaf nodes are objects.

Internal nodes are boolean operations.

CSG: Ray Tracing AlgorithmCSG: Ray Tracing AlgorithmSelect an eye point and a screen plane

FOR every pixel in the screen plane [num. pixels depends on image resolution]

FOR every leaf node in the CSG tree

compute and order by increasing depth the ray-object intersections.

ENDFOR

recurse left subtree returning intersection list [A]

recurse right subtree returning intersection list [B]

interleave sort A and B by increasing value along ray.

FOR all entries in intersection list

keep track of current state and use current intersection to access lookup table.

ENDFOR

RETURN new intersection list.

ENDFOR

A ~A B ~B

AB Cout Cout

A~B Cin Cin

~AB

~A~B

A Intersection B

CSG: Ray Tracing ExampleCSG: Ray Tracing Example

Constructive Modelling ExerciseConstructive Modelling Exercise

Specify the steps required for construction of the jug pictured below. Note: some stages can be achieved in several different ways. Enumerate them where possible.

Constructive Modelling SolutionConstructive Modelling Solution

1. Jug: Surface of Revolution

2. Handle: Extrude Ellipse

3. Spout: Free-Form Deformation

4. Combination: Union of Deformed Jug and Handle