Modelling

Outline

Modelling methods Editing models – adding detail Polygonal models Representing curves Patched surfaces

Introduction

There are many ways to model objects: none of these are right or wrong

Different modelling methods suit different objects and different people

Try some or all of these methods to see which suits you and the type of objects that you want to create

Modelling: types of model

Boundary modelling Polygonal models Extrusions and surfaces of revolution Patches

Procedural modelling Fractals Implicit surfaces

Volumetric models Constructive Solid Geometry (CSG) (which

you’ve already seen) Spatial subdivision

Editing models

Adding detail Adding/merging/deleting

points Booleans Transformations Scales

Polygonal models

As we’ve seen many computer models are made up of polygons, which in turn are made up of points, which are made up of 3 coordinates, x, y and z

Polygons can have any number of sides >= 3

When we model we should remember certain important points…

Polygons (cont)

When a computer renders our model it needs to know the angle of the surface to the light and the viewer

To do this it needs to calculate the normal to the surface, the vector that represents the direction that the surface is facing

This can be easy or hard depending on the nature of the polygon…

Polygons (cont)

Consider a four-sided polygon:

To calculate the normal, we can take the cross-product of two vectors representing two of the sides

This works because all the points in this example are coplanar: we would get the same normal from any

two sides But what if we raise one vertex?

Polygons (cont)

How do we calculate the normal now thatwe no longer have straight edges?

We can’t since the surface doesn’t point in one direction any more This can cause major problems for renderers Solution is to only use triangles, or be certain that your polygons are coplanar

Polygons (cont) The normal also defines the side of the polygon

that is visible: it is invisible from the other side This is because is has no thickness: i.e. a single

polygon could not really exist in the real world We can cheat and use double sided polygons,

i.e. they have a normal pointing out of both sides of them

This is generally bad practice and the mark of poor modelling!

Always think of how the object will be animated when at the modelling stage

Polygons (cont)

Another advantage of triangles is that they must be convex

This makes it easier to render again as the computer can calculate the inside and outside areas of the polygon.

Starting from primitives

Basic building blocks of many complex shapes are the primitives

These can be used to directly construct the shape

They can also be used as a cage for curved surface modelling

Adding detail to primitives

You can use other primitives to add or remove parts of your model (booleans)

You can directly add or remove (or merge) points

You can cut out areas using drills or stencils

You can slice through faces using a knife

Starting from points

Use Create Polygon Tool to draw polygons one by one

Very time consuming but gives ultimate control

Final model is very efficient (i.e. no wasted points)

Example: making a laptop Start from a primitive:

Modify the primitives Bevel the edges:

Modify the primitives Boolean subtract

Change some surfaces

Add some detail Boolean and bevel

Add some more detail

Create a hierarchy & animate

Why use anything else besides polygons?

Need to represent curves: very few objects that we need to model have straight edges

In most modelling programs the final model is approximated with polygons

Curves can be 2D or 3D There are many ways we can

represent curves…

Curve representation Line segments (polyline)

Simple set of points Awkward to edit Large number of points needed Never truly smooth

Splines Control points affect regions of curve

Easy to reshape

Truly curved

Compact and efficient to store

Actual curve is a mathematical representation

Usually cubic splines

Splines

Control points can affect curve in different ways

Two main categories Interpolating spline Approximating spline

Interpolating splines

Curve passes through control points Easy to place curve precisely Difficult to make completely smooth

E.g. Cardinal spline Passes through

all but first and last control point

Interpolating splines

Irregularly placed control points destroys continuity

Approximating splines

Curves passes near control points E.g. Bézier curves, B-Splines (‘Basis’

Splines) and NURBS (Non-Uniform Rational B-Splines)

Bézier curves

Four control points on minimal curve Uses tangent vectors to control

curvature

Bézier curves (2)

Equation defines the points on the curve:

1 and 0

between varies and curve theeffecting points

control theare ),( & ),( where

)1(3)1(3)1()(

)1(3)1(3)1()(

),(End),,(Start

222

31

3

222

31

3

2211

t

yxyx

yttyttytytty

xttxttxtxttx

yxyx

ddcc

dc

dc

Bézier curves (3) Features: Tangent at each end is equal to straight line

connecting the end point to the control point Moving a control point changes the entire curve to

the next control point Curve does not pass through control points Convex hull created by connecting the control points

contains the curve Distribution of points along curve is not uniform

Bézier curves (4)

Editing Bézier curves

Bézier curves (5) Breaking the

pair of tangent vectors

B-Splines

Generalisation of Bézier curve Curve doesn’t extend to first and last

control points Control points only influence local

part of curve

NURBS

Non-Uniform Rational B-Splines Curve passes through first and last

control points but not intermediate ones

Has ‘knots’ on the curve which can be moved as well as normal control points

Combines best features of interpolating and approximating splines

NURBS (2)

Features of curves All curves have a start and an end. This means we

can define points on the curve as t along curve from start.

A parameter defines how far along the curve the point is, so these are called parameterised curves

Can be used to define curved surfaces by moving the curve through space

Moving the spline along a path defined by a second spline creates a patch (or more accurately a bicubic patch if both splines are bicubic curves)

Type of patch depends on type of splines used to create it (e.g. a B-spline patch or Bézier patch)

This creates a network of control points, each of which can be moved individually

Curves in Maya: CV (control vertex)

Curves in Maya: EP (edit point)

Patched surfaces

Patched surfaces are networks of polygons in a regular array

The position of the polygons depends on a number of control points for the defining curves

Subdivision surfaces

Summary

Discussed modelling techniques Looked at a simple example Introduced curve representations

Cross-product

Given two vectors, A and B, the cross product is defined as:A B = Cwhere C is another vector calculated by:C = ( ya zb – yb za , za xb – zb xa , xa yb – xb

ya ) The length of C is given by:

|C| = |A||B|sin( )where is the angle between A and B

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