Curves

Representations of Geometry

Explicit: point clouds, polygon meshes, NURBS
Implicit: Level Sets, Algebraic surfaces

Definitions

Polylines

Polylines are sequence of vertices connected by straight line segments. Suitable for drawing simple shapes but lack any notion of smoothness required for modelling complex objects and surfaces.

A smooth curve is always converted to a polyline for rendering.

Curves

A curve can be seen as a continuous set of points in n dimensions. This definition allows us to easily generate points along a curve and visualise its shape.
A curve can be seen as a mapping from a interval S onto a plane. This definition allows us to describe trajectories and manipulate the speed at which we traverse the curve.$$ P : \mathbb{R} \ni S \mapsto \mathbb{R}^2 \hspace{2em} P(t) = \begin{pmatrix} x(t) \ y(t) \end{pmatrix} $$

Spline

A Spline is a curve that connects two or more points, based on control points. These control points determine the trajectory and shape of the spline.
Tessellation is the process of displaying the points on the curve $P (t)$ by discretising it at discrete $t$ values. This involves sampling the curve at intervals to create a series of points that approximate the curve.
Interpolation refers to the process of constructing a curve in such a way that it precisely passes through all points.
Approximation involves constructing a curve that doesn’t necessarily pass through all the points. Used to represent curves with few control points.

Differential Properties of Curves

🧠 Motivation:

Computing Normals of a Surface
Finding velocity of an animation
Analysing smoothness of a curve

Velocity can be computed using the derivative of the curve $P (t)$ w.r.t $t$ , and then normalised using the magnitude a unit vector.
Curvature/Acceleration is the derivative of the unit tangent. It helps us understand how quickly the curve is changing direction at each point.

Orders of Continuity

C0 Continuity: This simply means that the curve is continuous, with no abrupt jumps or gaps. However, there might be sharp kinks or corners at the seams where curves connect.
G1 Continuity: Geometric continuity ensures that tangents at the seams of connected curves point in the same direction.
C1 Continuity: This form of parametric continuity goes a step further ensuring that the tangents themselves are the same at the seams.
C2 Continuity and Higher: As we increase the order of continuity, the derivatives of the curves at the seams also match.

Types of Splines

1. Bezier Curves

A Bezier curve interpolates through some points (first and last) while approximating others. These curves need ( $n + 1$ ) control points, where $n$ is the degree of the curve
These curves are always bounded by the convex hull of its control points.
The tangents at the starting point and the endpoint; point in the direction of the line formed by their respective control points.
A cubic bezier curve in particular is represented by cubic polynomials. These four coefficients represent basis functions.

P (t) = (1 - t)^{3} P_{1} + 3 t (1 - t)^{2} P_{2} + 3 t^{2} (1 - t) P_{3} + t^{3} P_{4}

These four coefficients/basis functions are Bernstein polynomials.

Matrix Notation

P (t) = G B T (t) = Geometry \times Spline Basis \times Power Basis

P (t) = (\begin{matrix} x (t) \\ y (t) \end{matrix}) = (\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \end{matrix}) (\begin{matrix} 1 & - 3 & 3 & - 1 \\ 0 & 3 & - 6 & 3 \\ 0 & 0 & 3 & - 3 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ t \\ t^{2} \\ t^{3} \end{matrix})

(\begin{matrix} B_{1} (t) \\ B_{2} (t) \\ B_{3} (t) \\ B_{4} (t) \end{matrix}) = (\begin{matrix} 1 & - 3 & 3 & - 1 \\ 0 & 3 & - 6 & 3 \\ 0 & 0 & 3 & - 3 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ t \\ t^{2} \\ t^{3} \end{matrix})

(\begin{matrix} 1 \\ t \\ t^{2} \\ t^{3} \end{matrix}) = (\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 1 / 3 & 2 / 3 & 1 \\ 0 & 0 & 1 / 3 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix} B_{1} (t) \\ B_{2} (t) \\ B_{3} (t) \\ B_{4} (t) \end{matrix})

Bernstein functions are bounded in $[0, 1]$ .
Bernstein functions always sum to unity for any value of $t$ (Partition of Unity). This ensures that all curves are within the convex hull.
The basis functions are linearly independent, i.e. we can express any of the monomials as a linear combination of the others.
An essential characteristic of Bezier curves is that each control point influences the entire curve, not just its local vicinity. This means that adjusting any control point can significantly impact the entire curve’s shape.

Subdivision

Subdivision allows us to split a single Bezier curve into two separate curves.
De Casteljau Construction for Subdivision of Bezier Curves:
- Take the middle point of each of the 3 segments.
- Construct the two segments joining them.
- Join the middle point of those two segments
- Take the middle point.

2. B-Spline Curves

Need at least 4 control points.
B-Splines are locally cubic, meaning each segment of the curve is defined by a cubic polynomials.
Unlike Bezier curves, B-Spline curves are not constrained to pass through any control points, giving us more flexibility in shaping the curve.
B-Spline curves are also bounded by the convex hull of its control points.
Bezier curves also employ Bernstein polynomials (therefore partition of unity).

Conversion between Bezier and B-Spline curves

Bezier to B-Spline: Multiplying the bezier basis matrix by the inverse of the B-Spline basis matrix, obtaining new control points in the B-Spline basis.
B-Spline to Bezier: Multiplying the B-Spline basis matrix by the inverse of the Bezier basis matrix, obtaining new control points in the Bezier basis.

3. Non-Uniform Rational Basis Spline (NURBS)

Use homogenous coordinates, specifying weights for each control point known as knots in this context.
Additionally they require a knot vector to control the curve’s shape.
“Non-Uniform” refers to the varying spacing between the blending functions or knots in the curve.
“Rational” means the curve is defined by the ratio of cubic polynomials instead of just cubic terms.