Transcript CGA: Modelling

Modelling
Outline
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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 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
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)
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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:
Start  ( x1 , y1 ), End  ( x2 , y2 )
x(t )  (1  t ) 3 x1  t 3 x2  3t (1  t ) 2 xc  3t 2 (1  t ) xd
y (t )  (1  t ) 3 y1  t 3 y2  3t (1  t ) 2 yc  3t 2 (1  t ) yd
where ( xc , yc ) & ( xd , yd ) are the control
points effecting the curve and t varies between
0 and 1
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
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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 = ( 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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