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1
Intro
2
Examples of Paths
3
CHAPTER 1 - Bézier Curves
4
Lerp
5
Quadratic Bézier
6
Cubic Bézier
7
n-Degree Bézier
8
CHAPTER 2 - Bézier Splines
9
Cubic Bézier Spline
10
Spline Parameterization
11
The Anatomy of a Spline
12
Knot Values & Knot Intervals
13
Local Control
14
The flexibility of the Cubic Bézier
15
Tangent Points
16
CHAPTER 3 - Continuity
17
C⁰ Positional Continuity
18
Spline Derivatives
19
C¹ Velocity Continuity
20
C² Acceleration Continuity
21
Cascading Loss of Local Control
22
C³ Jolt Continuity
23
C∞ Continuity
24
Parametric Continuity Summary
25
CHAPTER 4 - Geometric Continuity
26
G¹ Tangent Continuity
27
G² Curvature Continuity
28
The Reflection Test
29
Seams in Circular Arcs
30
Angle Analysis
31
Curvature & The Osculating Circle
32
Curvature Combs
33
G¹ Algebraically
34
G² Algebraically
35
Geometric Continuity Summary
36
Continuity Summarized
37
Cusps
38
CHAPTER 5: The Extended Universe
39
Hermite Spline
40
Hermite ⇔ Bézier Conversion
41
Linear Spline
42
Cardinal Spline
43
Catmull-Rom Spline
44
Investigating Basis Functions
45
Basis Function Continuity Analysis
46
In Search of a C² Spline
47
B-Spline
48
Quiz Time!
49
Cubic Trajectory
50
Uniform Cubic Splines
51
Splines & Their Use Cases
52
Outro
53
NURBS
54
Color Spline
55
Credits & Freya Rambles
56
Cats helping me w. recording
Description:
Dive into an extensive 1-hour 14-minute video lecture on the continuity of splines, exploring various curve types and their applications in computer graphics and animation. Learn about Bézier curves, spline parameterization, and different levels of continuity, from positional to geometric. Discover the intricacies of Hermite splines, Cardinal splines, and B-splines, and understand their use cases in real-world scenarios. Gain insights into the mathematical foundations of these concepts through detailed explanations, visual demonstrations, and practical examples. By the end of this comprehensive tutorial, acquire a deep understanding of spline theory and its importance in creating smooth, continuous curves for various applications in computer graphics and design.

The Continuity of Splines

Freya Holmér
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#Mathematics #Computer Science #Computer Graphics #Bezier Curves
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Intro