The Continuity of Splines

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.

Intro
Examples of Paths
CHAPTER 1 - Bézier Curves
Lerp
Quadratic Bézier
Cubic Bézier
n-Degree Bézier
CHAPTER 2 - Bézier Splines
Cubic Bézier Spline
Spline Parameterization
The Anatomy of a Spline
Knot Values & Knot Intervals
Local Control
The flexibility of the Cubic Bézier
Tangent Points
CHAPTER 3 - Continuity
C⁰ Positional Continuity
Spline Derivatives
C¹ Velocity Continuity
C² Acceleration Continuity
Cascading Loss of Local Control
C³ Jolt Continuity
C∞ Continuity
Parametric Continuity Summary
CHAPTER 4 - Geometric Continuity
G¹ Tangent Continuity
G² Curvature Continuity
The Reflection Test
Seams in Circular Arcs
Angle Analysis
Curvature & The Osculating Circle
Curvature Combs
G¹ Algebraically
G² Algebraically
Geometric Continuity Summary
Continuity Summarized
Cusps
CHAPTER 5: The Extended Universe
Hermite Spline
Hermite ⇔ Bézier Conversion
Linear Spline
Cardinal Spline
Catmull-Rom Spline
Investigating Basis Functions
Basis Function Continuity Analysis
In Search of a C² Spline
B-Spline
Quiz Time!
Cubic Trajectory
Uniform Cubic Splines
Splines & Their Use Cases
Outro
NURBS
Color Spline
Credits & Freya Rambles
Cats helping me w. recording