3d curves

Question 1

Why are cubic polynomials commonly used in curve modelling?

Answer 1

They are the lowest-order polynomials that can provide smoothness (C¹ continuity) without excessive complexity.

Question 3

What does the parameter t represent in curve equations?

Answer 3

It represents the progression along the curve, typically ranging from 0 to 1.

Question 5

What is a general cubic polynomial curve equation?

Answer 5

p(t) = x₀ + t·x₁ + t²·x₂ + t³·x₃

Question 7

What is the matrix form of a cubic curve?

Answer 7

p(t) = C · Q(t), where Q(t) = [t³, t², t, 1]ᵀ

Question 9

What does the derivative of a cubic curve give?

Answer 9

The tangent vector, indicating direction and speed at time t.

Question 11

What is a Hermite curve defined by?

Answer 11

Two endpoints and the tangent vectors at each endpoint.

Question 13

How many constraints define a Hermite curve?

Answer 13

Four: p(0), p(1), p’(0), and p’(1).

Question 15

What is the general form of the Hermite curve equation?

Answer 15

p(t) = G · M · Q(t), where G is the geometry matrix and M is the Hermite basis.

Question 17

What are the Hermite basis matrix values?

Answer 17

[[2, -3, 0, 1], [-2, 3, 0, 0], [1, -2, 1, 0], [1, -1, 0, 0]]

Question 19

What is the expanded equation of the Hermite curve?

Answer 19

p(t) = p(0)(1-3t²+2t³) + p(1)(3t²-2t³) + p’(0)(t-2t²+t³) + p’(1)(-t²+t³)

Question 21

What does C⁰ continuity mean for curves?

Answer 21

The endpoints of adjacent segments match (position continuity).

Question 23

What does C¹ continuity mean for curves?

Answer 23

The tangents at the joining points match (smooth directional flow).

Question 25

What is a Bézier curve defined by?

Answer 25

Four control points: p₀, p₁, p₂, and p₃.

Question 27

Does a Bézier curve interpolate its control points?

Answer 27

Only the first and last control points (p₀ and p₃); not the interior ones.

Question 29

What is the tangent at the start of a Bézier curve?

Answer 29

p'(0) = 3(p₁ - p₀)

Question 31

What is the tangent at the end of a Bézier curve?

Answer 31

p'(1) = 3(p₃ - p₂)

Question 33

What is the Bézier curve equation?

Answer 33

p(t) = (1−t)³p₀ + 3t(1−t)²p₁ + 3t²(1−t)p₂ + t³p₃

Question 35

What geometric property do Bézier curves have?

Answer 35

The curve lies entirely within the convex hull of its control points.

Question 37

What is the Bézier blending matrix?

Answer 37

[[-1, 3, -3, 1], [3, -6, 3, 0], [-3, 3, 0, 0], [1, 0, 0, 0]]

Question 39

How can complex shapes be modelled with Bézier curves?

Answer 39

By joining multiple Bézier segments with C⁰ and optionally C¹ continuity.

Question 41

What is a Catmull–Rom spline?

Answer 41

A type of interpolating spline that passes through control points using automatically computed tangents.

Question 43

How are tangents computed in Catmull–Rom splines?

Answer 43

p'ₖ = (pₖ₊₁ - pₖ₋₁) / 2

Question 45

Does a Catmull–Rom spline require tangent input?

Answer 45

No, tangents are automatically computed from neighbouring points.

Question 47

What are the advantages of Hermite curves?

Answer 47

Direct control over endpoints and tangents; precise shape manipulation.

Question 49

What are the advantages of Bézier curves?

Answer 49

Smoothness, geometric intuitiveness, and control via 4 points.

Question 51

What is the main benefit of Catmull–Rom splines?

Answer 51

They interpolate the control points automatically with minimal setup.

Question 53

Which curves interpolate their endpoints?

Answer 53

Both Hermite and Bézier curves interpolate endpoints.

Question 55

Which curve type interpolates all control points?

Answer 55

Catmull–Rom splines.

Question 57

Why use blending matrices in curve modelling?

Answer 57

To separate geometry (control points) from parameter evaluation, making rendering efficient.

Question 59

What curve types are used in animation and path planning?

Answer 59

Hermite and Catmull–Rom curves due to their smoothness and control.