# Modelling. Outline  Modelling methods  Editing models – adding detail  Polygonal models  Representing curves  Patched surfaces.

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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 sidescross-product  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 that we 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

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:

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 = C where C is another vector calculated by: C = ( y a z b – y b z a, z a x b – z b x a, x a y b – x b y a )  The length of C is given by: |C| = |A||B|sin(  ) where  is the angle between A and B Back…

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