surfaces

Question 1

What are the three main surface representations in 3D?

Answer 1

Explicit, parametric, and implicit.

Question 3

What is the explicit form of a surface?

Answer 3

x₃ = f(x₁, x₂)

Question 5

What is a key limitation of explicit surface representation?

Answer 5

It cannot represent vertical surfaces or closed shapes like spheres.

Question 7

What is a parametric surface?

Answer 7

A surface defined as f(u, v) = [x₁(u,v), x₂(u,v), x₃(u,v)], using two parameters.

Question 9

What is the parametric form of a sphere?

Answer 9

f(θ, φ) = r · [sinφ·cosθ, sinφ·sinθ, cosφ]

Question 11

What is an implicit surface?

Answer 11

A surface defined as the set of points where f(x₁, x₂, x₃) = 0.

Question 13

What does f(x) < 0 mean in an implicit surface?

Answer 13

The point is inside the surface.

Question 15

What does f(x) > 0 mean in an implicit surface?

Answer 15

The point is outside the surface.

Question 17

What is the implicit equation of a sphere of radius r?

Answer 17

f(x) = x₁² + x₂² + x₃² - r² = 0

Question 19

How can you model complex surfaces using implicit functions?

Answer 19

By combining simple shapes using min (union) or max (intersection) operations.

Question 21

What is the union operation in implicit modelling?

Answer 21

f(x) = min(f₁(x), f₂(x))

Question 23

What is the intersection operation in implicit modelling?

Answer 23

f(x) = max(f₁(x), f₂(x))

Question 25

What is the goal of marching squares?

Answer 25

To extract 2D contour lines from implicit functions over a grid.

Question 27

How many marching squares configurations exist?

Answer 27

16, grouped into 4 classes by symmetry.

Question 29

What is the main ambiguity in marching squares?

Answer 29

When opposite corners of a square are inside and outside the shape, creating uncertain center classification.

Question 31

What is marching cubes?

Answer 31

A 3D algorithm to extract surfaces from volumetric data using implicit functions.

Question 33

What are the steps of marching cubes?

Answer 33

1) Load voxel layers, 2) Classify cube corners, 3) Build case index, 4) Interpolate edges, 5) Create triangles.

Question 35

How many possible cases are there in marching cubes?

Answer 35

256 (based on 8-bit cube corner configurations).

Question 37

What is the output of marching cubes?

Answer 37

A triangle mesh approximating the implicit surface.

Question 39

What does increasing voxel resolution do in marching cubes?

Answer 39

Improves surface detail and accuracy.

Question 41

What are subdivision surfaces?

Answer 41

Meshes refined iteratively to create smooth, continuous surfaces.

Question 43

What are the two types of subdivision schemes?

Answer 43

Approximating and interpolating.

Question 45

What is an approximating subdivision scheme?

Answer 45

A scheme that replaces old points and creates new ones (e.g., Chaikin’s algorithm).

Question 47

What is an interpolating subdivision scheme?

Answer 47

A scheme that retains original control points and adds new ones (e.g., 4-point method).

Question 49

What is Chaikin’s corner-cutting algorithm?

Answer 49

An approximating subdivision that creates smooth curves by inserting points in 1:3 ratio on edges.

Question 51

What does subdivision achieve after many iterations?

Answer 51

A smooth limit curve or surface.

Question 53

What are key properties of good subdivision algorithms?

Answer 53

Convergence (stability) and smoothness (continuity).

Question 55

Where are subdivision surfaces commonly used?

Answer 55

In animation (e.g., Pixar), CAD modeling, and computer graphics.

Question 57

What is the main advantage of parametric surfaces?

Answer 57

They allow easy traversal and animation over the surface using u, v parameters.

Question 59

What does an implicit surface allow you to easily test?

Answer 59

Whether a point is inside, on, or outside the surface.

Question 61

What does marching cubes output that is usable in rendering and printing?

Answer 61

A triangle mesh approximating the implicit surface.

Question 63

Why use subdivision surfaces in design?

Answer 63

They give smooth, controllable shapes from coarse polygon meshes.