Computer Graphics
COMP3421/9415 2021 Term 3 Lecture 8
What did we learn last lecture?
Scene Graphs
● Organisation for complicated scenes and hierarchical objects Depth Testing
● Rendering things in the right order
● Seeing only what’s in front
Blending
● Also seeing what’s behind something if it’s transparent!
What are we covering today?
Parametric Equations
● Linear Interpolation
● Using parameters to control movement and curves
● Also using time as the driver
Linear Interpolation
Maths inside Fragment Shaders
How do we choose a texture coordinate or colour in between two vertices?
● Vertices A and B have texture coordinates AT and BT
● What’s the texture coord of my fragment F?
● If it’s halfway between them:
○ FT = AT * 0.5 + BT * 0.5
○ This means both A and B have 50% influence over the
texture coordinate
AFB
Linear Interpolation
This works for all points between the two vertices
● Not just the halfway point
● If F is at A:
○ FT = AT * 1.0 + BT * 0.0 ● Or at B:
○ FT = AT * 0.0 + BT * 1.0
● You can see a pattern forming:
○ FT = AT * (1.0 – t) + BT * t
● If we use a parameter, t, we can represent all the possible values
between the two points
What can we do with this technique?
Finding points between two points
● Moving t between 0 and 1 can give us any point between A and B
● This gives us a very simple way of moving between vertices
● We can describe a line or path by only using the end points
● We can also do a smooth transition between properties like colours or
texture coordinates
Using Time as a Parameter
Moving in time
● If we change our t based on delta time, we can animate movement between positions
● We can apply an interpolation to a coordinate or vector, so we could do something like a smooth delta time based pan of a camera
t = elapsed time/5sec
Change the camera’s “target” by linear interpolation and recalculate lookAt each frame. In 5 seconds we’ll pan from A to B.
Parametric Equations
Straight Lines Only?
Let’s add more interesting paths
● Linear Interpolation is straight lines between two points or values
● But the idea of parametric equations can do way more than that
● Try this one:
○ x = cos(t), y = sin(t)
○ This one even works for any value of t
Control Points
Points that influence a line based on a parameter
● We’ve seen a single line with two points
● What about more points?
● Let’s look at Bezier Curves
● They’re parametric
● And use multiple points
Bezier Curves/Splines
Makes use of Linear Interpolation
● If we have multiple points, we’ll linearly interpolate at multiple levels
● Each line we draw is a tangent to the curve
Different types of Bezier Curves/Splines
Each curve has a parametric formula
● Two points (Linear Interpolation) ○ P = (1-t)P1 + tP2
● Three points (Quadratic Curve)
○ P = (1-t)2P1 + 2(1-t)tP2 + t2P3
● Four points (most commonly used in Graphics) ○ P = (1−t)3P1 + 3(1−t)2tP2 + 3(1−t)t2P3 + t3P4
Useful Properties
Bezier curves . . .
● Tangents based on control points
○ At either end, the two closest control points form a tangent
● Join multiple curves together smoothly using colinear control points
More Advanced Curves and Splines
We can join them together, but . . .
● While the gradient will be equal at the join
● No guarantees about the second derivative
Why is that an issue?
● If we’re using this curve as a path to move on, the speed won’t be the same at the join
● Check out B-Splines if you want to know more about possible solutions
Splines in Graphics
● Polygon Rendering works in straight lines
○ Want a lot of points spaced out along a curve?
○ A parametric curve allows us to create an arbitrary number of points
○ We can draw our lines between those points to approximate the curve
● If we are moving an object along a curve
○ We can reach arbitrary positions using our parameter (link delta time to the parameter)
○ If we need a tangent, we can do a simple approximation by creating another nearby point
● Easy to modify
○ Just move control points around to change the nature of the curve
● A downside: Can’t quite control size and speed
○ You can’t necessarily move along a spline at a fixed speed
○ Parameter based movement is based on how far apart control points are
Break Time
‘s interesting career
● Invented Texture Mapping (1974)
● Invented the Catmull Rom Spline (1974)
● Used this and similar techniques in Keyframe
Animation (1970s)
● Ended up at Industrial Light and Magic (owned by
) (1979)
● buys Lucasfilm digital division and
creates Pixar (1986)
● was at one point President of Disney
and Pixar
Image credit: (VES Awards 2010)
Using Splines
Hello Teapot our old friend
How would we create this object?
● Several tricks in use here!
● We have curves, but how are we creating
surfaces?
● We can create something like the teapot
with just some simple equations and transform matrices
Image credit: School of Computing, University of Utah
Surfaces of Revolution
The body of the teapot
● We can create a curve (side of the pot)
● Translate that curve away from (0,0,0)
● Create a series of points along that curve (using values of t)
● Then we can rotate the curve and its points around the Y axis
● At different rotation angles (the Utah teapot has about 30) we can create
vertices
● Make triangles from those vertices and build up buffers
A Surface of Revolution
In Images
Create a curve
Translate away from y axis Choose t values for points
A Surface of Revolution
In Images (continued)
Rotate the curve to different Between two of the close curves, orientations around the y axis create vertices and triangles
Extrusion
Dragging a shape through space, the handle of the teapot
● We can create a curve (the centre of the handle)
● Translate that curve into the correct position at the back of the teapot
● Create a series of points along that curve (using values of t)
● Create a circle (the thickness of the handle)
● Place the circle at each those points
● Use the circle to create vertices
Extrusion
In Images
Create a curve Translate the handle into position
Extrusion
In Images (continued)
Create a circle and place it at Create vertices around the circles different points along the curve and create triangles between them
Other parts
The lid
● Another surface of revolution, just a slightly more complex curve The spout
● This gets harder
● An extrusion that is scaled based on distance from the pot?
Techniques for 3D Object Creation
Digital Artists might use a lot of these
● Rotation and extrusion are definitely used
● And simple things like scale, rotate and translate!
● Artists will not usually be expected to be computer scientists and
mathematicians
○ So there are tools like Maya and Blender to hide the details
● There’s also raw sculpting though
○ Digital Clay! (Zbrush uses this kind of technique)
Let’s guess how some things were made
Toy Story is a classic that’s historically very important!
● Potato heads
○ Obvious separate objects with scene graph
attachments
○ Some simple rotational volumes
● Buzz Lightyear
○ Transparency nightmare!
○ Very simple scene graph with rigid components
● Woody
○ Squishy bits . . . we’ll talk about these later!
Images credit: Disney Pixar
What did we learn today?
Parametric Maths and its applications
● Linear Interpolation
● Parametric Curves/Splines
● Using Bezier Curves in 3D
● A small look at how 3D Artists create some game/film assets
Homework
● Watch Toy Story and Monsters, Inc. (Two Pixar films 6 years apart)
● See if you can guess how some things were made and also the technical
advancements between the two movies