SURFACES (W2)

Question 1

What are the 4 ways to define a surface?

Answer 1

1) Explicit: mesh, point
2) Tensor product splines
3) Implicit: formula
4) Procedural surfaces

Question 2

What is a mesh representation

Answer 2

Either a triangle or a polygon mesh
Each polyline or face is defined by a set of vertices
Can subdivide to make the surface smoother

Question 3

Defining a surface using bezier and b-spline curves

Answer 3

If we have multiple bezier curves vertically eg at v=0, v=1, …v=n
We can build a surface by defining the surface horizontally with more splines
So we are reusing control points to define multiple curves

Question 4

Surface formula

Answer 4

Can be defined in the same way as a curve
P(u, v) = (1-u)^3 P1(v)
+ 3u(1-u)^2 P2 (v)
+ 3u^2(1-u) P3 (v)
+ u^3 P4 (v)

The v tells us where along the surface we put that bezier curve

Question 5

What is a bicubic bezier surface

Answer 5

When the surface is built both horizontally and vertically using bezier curves
We can always keep adding more
-The control points are reused so we are saving space

Question 6

What is a parametric surface

Answer 6

Interchangeable with “Tensor Product splines”
A more general term for when we can define our surface using parameters
Include bicubic bezier surfaces

Question 7

Derivatives of a specific point P on a 3D surface

Answer 7

We can use v (more vertical) and u as the axis of the surface
δP / δu
δP / δv
We can go in either v or u direction and find that velocity (T or B)
We can also compute the normal using the 2nd derivative
We can find the binormal (B) by finding the cross-product of the Tangent and Normal

Question 8

Adv and Disadv of Tensor Product splines

Answer 8

Adv
Takes up less storage space - only needs control points
Based on NURBS, very good for rendering

Disadv
Harder to render as they are often converted to triangles by the GPU at the end anyway
Harder to define patch boundaries

Question 9

What do we call a small set of a surface

Answer 9

A Patch

Question 10

Defining a patch

Answer 10

We can define a patch P using u and v
P(u,v)
At a specific point x
Px(u,v) =
[B1(u),…B4(u)] (1x4 matrix).
[Px1,1….Px4,4] (4x4 matrix).
[B1(v),…(B4(v)] (4x1 matrix)

So now we have to Bernstein functions, one in u direction and one in v direction
Px are the control points

Question 11

Patches and curves In Matrix form:

Answer 11

Curves:
P(t) = GBT(t)

Patches:
Px(u,v) = T(u)^T. B^T. G^x.B.T(v)

G is now specifically defined at point x
Add B transpose as we need both directions now (u and v)
And same for T transpose

Question 12

What do we call the technique used to paint dimensionality on a surface

Answer 12

Displacement
When we do not want a surface to be smooth
Eg the whiter we make a pixel, the more it will travel up in the normal when modelled

Question 13

What is tesselation

Answer 13

Or subdivision
To make the surface smooth again once we have displaced the points

Question 14

What is Chaikin’s Algorithm

Answer 14

“corner-cutting”
A subdivision technique
We start with “the control polygon” with vertices called “the control points”
Then subdivide further and further until we reach a smooth line or surface called “the limit”

Produces a quadratic B-spline curve

Question 15

What are Implicit surfaces(adv and disadv)

Answer 15

We are given a formula and can define a surface in a 3D space
Adv
Very efficient if we want to a check whether a point is inside or outside, we just have to substitute x,y,z values into the formula
Efficient for boolean operations
Can handle weird topology, eg holes

Disadv
It cannot easily generate points on the surfaces eg a bump
Hard to define asymmetric geometry

Question 16

Working out if a point is inside or outside a curve

Answer 16

f(x,y) = 0 : on the curve
f(x,y) < 0 : inside the curve
f(x,y) > 0 : outside the curve

Question 17

Surfaces of revolution (procedural surfaces)

Answer 17

Uses a base shape then revolves it around an axis
S(u,v) = R(v)q(u)
Where R is a rotation matrix that only travels in v direction
and q is a line in the u direction

Question 18

Generalised cylinders (procedural surfaces)

Answer 18

Given a 2d profile and a 3d curve
Sweep the profile along the 3d curve
S(u,v) = M(c(v))q(u))
c is the trajectory
q is the profile curve itself
M is the rotation

Surface of revolution can be seen as a special case of a generalised curve when the trajectory (c) is a circle

Question 19

Frenet Frame

Answer 19

Is defined by the first derivative of a point on a surface (Tangent - giving us the velocity)
Also have the second and third derivatives of the curve

Question 20

Frenet frames: Inflection problem

Answer 20

When the curve face flips so the sign changes

Question 21

Point set

Answer 21

Just a noisy point cloud
Not connected to anything
Can define a reasonable surface
Application: laser range outputs a point set
Can be very favourable in modelling for implicit surfaces and points sets