15 - 16 - Parametric Representations

Question 1

What is a parametric representation

Answer 1

involves modelling curves on the surfaces of objects.

Representing a point mathematical object by parameters in a lower dimension.

curve = t
the surface can be described by (u,v)

Question 2

Formula to find the parametric equation for a line joining 2 points

Answer 2

x = xa+ (xb-xa)t
y = ya + (yb-ya)t

then find value of t (distance along the line)

ex point (1,4) and (5, -8) value of x when y = 0

x = 1+(5-1)t
y = 4+(-8-4)t

0 = y
0 = 4 - 12t
t = 1/3

Substitute t into the equation for x:
x = 5/3

Question 3

Bezier Curve

Answer 3

Bezier curves are polynomial curves defined by control vertices.

A Bezier curve has the following properties:
* Usually cubic
* Passes through the first and end control points
* The curve lies within the convex hull of the control points
* The curve Q(t) is expressed as a summation of the basis functions

Question 4

Joining Bezier Curves

Answer 4

joined by co-locating end control points.

Smoothness can be maintained by keeping end pairs of control point co-linear.

Question 5

There are 3 types of continuity:

Answer 5

• Zeroth order – Co Segments meet
• First order – C1 continuity
  o Segments meet
  o Tangents are equal (first derivatives are equal)
• Second order – C2 continuity
  o Segments meet
  o C1 continuity
  o Second derivates are equal – curvature is much smoother

Question 6

Bezier Surfaces Issues

Answer 6

Control is non local, moving control points in one area will affect other control points

Question 7

De Casteljau’s Algorithm

Answer 7

is a recursive algorithm used to evaluate a Bezier curve.

The algorithm works by iteratively dividing the curve into smaller pieces and evaluating the position of the curve at a specific parameter value.

Question 8

Cubic Splines

Answer 8

Cubic splines are a type of interpolation curve used to approximate complex curves with a series of simpler curves.

They are an alternative to Bezier curves/surfaces.

There are two types of splines:
* Approximating splines
* Interpolating splines

Question 9

Subdivision

Answer 9

technique used to generate smooth surfaces from a relatively coarse set of control points.

The basic idea behind subdivision is to iteratively refine a surface by subdividing each polygonal face into smaller faces and adjusting the positions of the control points based on some interpolation or averaging scheme.

Question 10

Subdivision Curves

Answer 10

• C1 Continuous Curves (Chaikin)
• C2 Continuous Curves similar to chaikin , instead of using points at ¼ and ¾, midpoints and adjusted vertices are used

Question 11

Subdivision Faces

Answer 11

successor to NURBs patches; they are easy to manipulate and not confined to quadrilateral meshes. However, they require a lot of memory.

Question 12

Catmull-Clark Subdivision

Answer 12

surfaces are widely used for modelling complex shapes, such as characters, creatures, and architectural forms.

They are especially useful for creating organic shapes that require smooth, continuous surfaces with complex topologies – they have C2 continuity

Question 13

Displacement Mapping

Answer 13

a texture map is used to modify the positions of the vertices in a mesh, creating a more detailed and complex surface.

The texture map typically contains information about the height or depth of the surface at each point, which is used to modify the position of the vertices along their normals.

Produces true shadows unlike bump mapping

Question 14

Tessellation

Answer 14

used to offload refinement of triangles into more vertices from the CPU to the GPU.

2 tessellation shaders:
* Tessellation Control Shaders (TCS)
* The tessellation primitive generator takes the input patch and subdivides it based on values computed by the TCS
* Tessellation Evaluation Shaders (TES)

Question 15

Tessellation Control Shaders (TCS)

Answer 15

How much tessellation to do

Question 16

Tessellation Evaluation Shaders (TES)

Answer 16

takes the tessellated patch and computes the vertex values (and new edges/faces) for each generated vertex

Question 17

Tessellation Levels

Answer 17

defines how many segments an edge is tessellated into

for example, a tessellation level of 4 means that an edge will become 4 edges (segments) (2 vertices become 5 vertices ):

o——————o
o—o—o—o—o

Question 18

Inner Tessallation

Answer 18

tessellation within each patch

Question 19

outer tessallation

Answer 19

defines how many segments an edge is tessellated to (outer level of 5 means 1 edge becomes 5 edges)