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COMP37111 > CURVES (W1) > Flashcards

CURVES (W1) Flashcards

1
Q

What are the 3 main stages of the Graphics Pipeline?

A

Modelling
Animation
Rendering

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2
Q

What are the two types of geometry

A

Explicit
Implicit

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3
Q

What are the 3 types of explicit geometry we learn about

A

Direct points
Polygon mesh
Bezier Curve

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4
Q

What are 2 types of implicit geometry we learn about

A

Level Sets
Algebraic surfaces
(Neither of these describes explicit points)

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5
Q

What are polylines

A

Sequence of vertices connected by straight line segments
Lack smoothness
Important because they are essentially what a GPU draws at the end

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6
Q

What is a Spline

A

The general term for a curve that gracefully connects 2 or more points

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7
Q

What is interpolation

A

A way of constructing a curve so it passes through every points
Ideal when we want to preserve given data points

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8
Q

2 ways of defining a curve

A

A continuous 1d set of points
-easy to edit

Treat a curve as a mapping from an interval S onto a plane
P: S ∈ R -> R^2
P(t) = (x(t), y(t))

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9
Q

What is approximation

A

Smoothly approximates the overall shape
More flexible
Better for curves with fewer control points

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10
Q

What are the 3 main types of Spline

A

Cubic Bezier Curve
Cubic B-Spline Curve
NURBS: Non-uniform Rational Basis Spline

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11
Q

What degree is a cubic bezier curve

A

Degree 3

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12
Q

What is a cubic bezier curve

A

A curve which connects p1 and p4 and interpolates between p2 and p3

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13
Q

How does curve degree relate to control points

A

Degree = control point -1

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14
Q

What are the 2 geometrical properties of the cubic bezier curve

A

Forms a convex hull defined by the control points, the curve is always inside this bounded box
At the 2 end points, both points are tangent to curve

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15
Q

What are the 4 coefficients of each point in P(t) called

A

Bernstein Polynomials
P(t) =
(1-t)^3 P1
+ 3t(1-t)^2 P2
+ 3t^2(1-t) P3
+ t^3 P4

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16
Q

What are the two key properties of the cubic bezier curve Bernstein polynomials

A

non-negativity: 0 < all values < 1
Unity sum: the sum of all the Bernstein polynomials is always equal to 1
- this is why the curve always remains inside the convex hull

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17
Q

Matrix Notation for cubic bezier curves

A

P(t) = GBT(t)
= Geometry X Spline Basis X Power Basis
Where the Geometry is the 4 control points ((x1,y1), (x2,y2), (x3,y3), (x4,y4))
The Spline Basis is the matrix of Bernstein coefficients
And the Power Basis is the matrix of t powers (1, t, t^2, t^3)

18
Q

Calculating the Bernstein function for any bezier curve

A

There is a formula (does not need to be memorised)
Bni(t) = n! / i! (n-i)! ti(1-t)^(n-i)

19
Q

What is the benefit of using matrix notation for bezier curves

A

Can use B-1 (Spline basis inverse) to convert and finding the unknown matrix

20
Q

What is subdivision

A

A technique which allows us to split a single curve into two sub curves
These resulting curves are also cubic bezier curves (still with 4 control points)
Particularly useful for adding finer detail to the two curves
And reducing complexity if we have higher degree curves

21
Q

What is De Casteljau Construction

A

A technique for subdividing bezier curves
Essentially find the midpoints of the sides of the convex hull
Connect the midpoints and find the midpoint of that
The midpoints of this final connecting line is the midpoint of the curve
We also do not need to split into half, we can divide by any arbitrary t

22
Q

What is the purpose of computing differential properties of curves

A

It allows us to find the normal of a surface, or a velocity of an animation or the smoothness of a curve

23
Q

How do we find Velocity

A

The 1st derivative of the curve P(t)
It is the tangent of the curve at point t

Tangent at point t:
T(t) = P’(t) / ||P’(t)||

||P’(t)|| is just the normalization term to make the tangent a unit vector
This T is the physical property of the velocity

24
Q

How do we find Curvature

A

(same as acceleration)
The second derivative the curve P(t)
Or first derivative of T(t)
The change in speed

Normal at point(t)
N(t) = T’(t) / ||T’(t)||

Curve normal gives us the curvature (or acceleration)

25
Q

What are the Orders of Continuity

A

Important when we want to combine multiple curves
Different orders describe the smoothness, kinks or breaks

26
Q

What is C0 Continuity

A

The minimum requirement for continuity
Simply means 2 curves are connected
ω shape

27
Q

What is G1 Continuity

A

The tangents at that point are pointing towards the same direction but not necessarily the same
-> 2 shape

Visually pleasing and sufficient for modeling surfaces and objects

28
Q

What is C1 Continuity

A

Ensures the tangents themselves are the same at that point

29
Q

What is C2+ Continuity

A

Ensures not only the tangents are the same but also the tangent derivatives

30
Q

How do we find the continuity of a curve

A

Calculate the tangent and the normal and compare with definitions for C0, G0, C1, etc.

31
Q

What is a Cubic B-Spline Curve

A

Unique in that in needs at least 4 control points to be defined
“Locally cubic” Each segment of the curve is defined by a cubic polynomial
If we chain these together it is still a cubic b-spline curve (even though there are many control points)
Never pass through any of the control points
-Gives us more control

32
Q

What is the degree of the Cubic B-Spline Curve

A

degree 3

33
Q

Does a B-spline curve also have a convex hull

A

Yes the curve is still bounded by its control points with a convex hull

34
Q

Key properties of the B-Spline curve bernstein polynomials

A

Unity sum: always adds to 1
Pick any arbitrary t - all coefficients adds to 1
so it always stays within its convex hull

35
Q

Finding a point P at t

A

Subsitute t into the formulae for B1(t), B2(t), B3(t), B4(t) Add them together to give point P(t)

36
Q

Converting between Bezier and B-Spline

A

Let B1 = Bezier and B2 = B-Spline
P(t) = GBT(t)
P(t) = G1.B1.T(t)
As T(t) does not change for B1 or B2
P(t) = G1.B1.B2^(1).B2. T(t)
So
G2 = G1.B1.B2^(1)

37
Q

How can we use converting between bezier and B-Spline

A

Given a new set of control points, we can define the same curve using either Bezier or B-Spline depending on our needs

38
Q

What is NURBS

A

Non-uniform Rational Basis Spline

39
Q

How are NURBS created

A

Employ homogeneous control points by adding weights to each control point
This allows us to control the flow of the curve
Most 3D modelling uses NURBS
Allows a wide range of smooth curve modelling

40
Q

What does Non-uniform in NURBS mean

A

varying spacing between the blending function
Due to different weight assigned to control points

41
Q

What are control points called in NURBS

A

knots

42
Q

What does Rational mean in NURBS

A

The curve is defined by the ratio of the cubic polynomials instead of just the cubic term

COMP37111 (17 decks)

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