2d curves

Question 1

What is a curve in visual computing?

Answer 1

A continuous 1D shape defined in 2D or 3D space using mathematical functions.

Question 3

What are the three main ways to represent a 2D curve?

Answer 3

Explicit, parametric, and implicit representations.

Question 5

What is an explicit curve representation?

Answer 5

A representation where one variable is a function of the other, e.g., x₂ = f(x₁).

Question 7

What is a key limitation of explicit curve representation?

Answer 7

It cannot represent vertical or multi-valued curves.

Question 9

What is a parametric curve representation?

Answer 9

A curve defined by a parameter t that generates both x₁ and x₂ coordinates.

Question 11

What is an example of a parametric equation for a circle?

Answer 11

[x₁, x₂] = r · [cos(t), sin(t)]

Question 13

Why are parametric curves useful in animation and design?

Answer 13

They are easy to iterate over and can represent curves with direction and orientation.

Question 15

What is an implicit curve representation?

Answer 15

A set of points that satisfy an equation f(x₁, x₂) = 0.

Question 17

What is an example of an implicit curve?

Answer 17

A circle: x₁² + x₂² - r² = 0

Question 19

What is a benefit of implicit representation?

Answer 19

It allows easy point-in-region tests (e.g., to check if a point lies inside a shape).

Question 21

What is the disadvantage of using many straight lines to model curves?

Answer 21

It is inefficient and results in unsmooth approximations.

Question 23

What is a better method to model complex curves?

Answer 23

Using smooth, piecewise polynomial curves.

Question 25

What is a piecewise curve?

Answer 25

A curve made of multiple segments, each defined by a polynomial function.

Question 27

What is C⁰ continuity in curves?

Answer 27

Position continuity — the endpoints of two segments meet.

Question 29

What is the condition for C⁰ continuity?

Answer 29

f(1) = g(0), where f and g are adjacent segments.

Question 31

What is C¹ continuity in curves?

Answer 31

Tangent continuity — the first derivatives of the segments match at the join.

Question 33

What is the condition for C¹ continuity?

Answer 33

f′(1) = g′(0), meaning the curve joins smoothly.

Question 35

What does C¹ continuity imply about C⁰?

Answer 35

C¹ continuity also guarantees C⁰ continuity.

Question 37

What type of polynomial is often used for curve segments?

Answer 37

Cubic polynomials (degree 3).

Question 39

What is the general form of a polynomial curve segment?

Answer 39

f(t) = Σ t^i · xᵢ, for i = 0 to n

Question 41

What are Hermite curves defined by?

Answer 41

A start point, end point, and tangents at both ends.

Question 43

Why are Hermite curves useful?

Answer 43

They ensure smooth transitions and are useful in animation and path planning.

Question 45

What is the key benefit of polynomial curve modelling?

Answer 45

It enables precise control over shape, smoothness, and continuity.

Question 47

What are the advantages of parametric curves over explicit ones?

Answer 47

They can handle multi-valued curves, vertical segments, and allow for direction and motion control.

Question 49

What’s the main reason to use piecewise polynomials in curve design?

Answer 49

They allow complex, smooth shapes while maintaining control at segment boundaries.