Skip to content

Curves

Path of a continuously moving point in the space. Set of all points where the moving point has been.

Mathematical Description

  • Parametric Curves:
    • In 2D: \(C(t) = (x(t), y(t)), C: \mathbb{R} \rightarrow \mathbb{R^2}\)
    • In 3D: \(C(t) = (x(t), y(t), z(t)), C: \mathbb{R} \rightarrow \mathbb{R^3}\)
  • Implicit Curves: Defined as the set of points satisfying an equation \(f(x, y) = 0\).
    • In 2D: \(C = \{(x, y): f(x, y) = 0 \}\)
    • In 3D: \(C = \{(x, y, z): f(x, y, z) = 0 \}\)

Warning

Always ensure that Implicit curves are of the form \(f(x, y) = 0\) and not \(f(x, y) = c\).

Curve Parametric Implicit
Line \(p(t) = p_0 + (p_1 - p_0)t, t \in (-\infty, \infty)\) \(C = \{(x, y): ax + by + c = 0\}\)
Circle \(p(t) = (r\cos(t), r\sin(t)), t \in [0, 2\pi]\) \(C = \{(x, y): x^2 + y^2 - r^2 = 0\}\)

Note

Always mention the range of the parameter \(t\) when defining a parametric curve.

Parametric Polynomial Curves: \(C(t) = \{x(t), y(t)\}\) is polynomial iff \(x(t)\) and \(y(t)\) are polynomials in \(t\).

Approximation Curves

Do not need to interpolate all the points(i.e. pass through all the points). Approximation curves are used to represent the shape of the curve.

Examples:

  • Bezier Curves
  • B-Spline Curves
  • Catmull-Rom Splines
  • etc.

Bezier Curves

Quadratic Bezier Curve
Quadratic Bezier Curve

Bezier curves are defined by a set of control points. Nth order Bezier curve is defined by N+1 control points. Equation of Linear Bezier curve is given by:

\[B^1(t) = (1-t)P_0 + tP_1\]

where \(P_0\) and \(P_1\) are the control points. Bezier curves are defined recursively. The general equation for Nth order Bezier curve is given by:

\[B^n(t) = (1-t)B^{n-1}(t) + tB^{n-1}(t)\]

or they can be defined using all the control points as:

\[B^n(t) = \sum_{i=0}^{n} {n \choose i} (1-t)^{n-1}t^i P_i\]

where \({n \choose i}\) is the binomial coefficient, and \(P_i\) are the control points.

Properties of Bezier Curves

  • The curve always passes through the end points, i.e. \(B^n(0) = P_0\) and \(B^n(1) = P_n\).
  • The curve is a straight line iff all the control points are collinear.

  • The start and end tangent vectors are parallel to the line joining the first and last control points, i.e.

    \[\frac{dB}{dt}|_{t = 0} \propto \overrightarrow{P_0 P_1} \hspace{40px} \text{ and } \hspace{40px} \frac{dB}{dt}|_{t = 1} \propto \overrightarrow{P_{n-1} P_n}\]

  • Convex Hull Property: The curve lies within the convex hull of the control points.

Convex Hull
Convex Hull (the boundary is P0 P1 P2 P3)

Note

Convex Hull is the smallest convex set that contains all the control points. Convex set is a set where the line segment joining any two points in the set lies entirely within the set.

  • Each \(n\)th order Bezier curve has an equally shaped \(n+1\)th order Bezier curve
  • Is Affine Invariant: The curve remains the same under affine transformations (i.e. translation, rotation, scaling, and shear).
Rational Bezier Curves

Rational Bezier curves are defined by dividing each control point by a weight. The equation for Rational Bezier curve is given by:

\[B^n(t) = \frac{\sum_{i=0}^{n} {n \choose i} (1-t)^{n-1}t^i w_i P_i}{\sum_{i=0}^{n} {n \choose i} (1-t)^{n-1}t^i w_i}\]

where \(w_i\) are the weights. They have better local control and can represent conic sections, but they are computationally expensive.

Interpolation Curves

Given a set of points \(P_0, P_1, \dots, P_n\) , we want to construct a curve that passes through all the points. \(C(k) = p_k\) where \(k = 0, 1, \dots, n-1\).

Some interpolation curves are:

  • Lagrange Interpolation
  • Piecewise Bezier Curves
  • Piecewise B-Spline Curves
  • etc.

Original Data Constant Interpolation Linear Interpolation Polynomial Interpolation

Different types of interpolation

Lagrange Interpolation

Given a set of points \(P_0, P_1, \dots, P_n\), the Lagrange interpolation polynomial is given by:

\[L(k) = \sum_{i=0}^{n} P_i l_i(k)\]

where \(l_i(k)\) are the Lagrange basis functions given by:

\[l_i(k) = \prod_{j=0, j \neq i}^{n} \frac{k - k_j}{k_i - k_j}\]

We rarely use Lagrange interpolation in practice because of the huge oscillations and large interpolation error.

Piecewise Interpolation Curves

Known as "Poly-Curves". Each segment between two interpolation points is represented by a curve.

Linear Piecewise Interpolation Piecewise Cubic Bezier Interpolation

Two types of interpolation, Linear on the right and Cubic Bezier on the left

Continuity in Piecewise Curves

Parametric Continuity \(C^n\)

  • Segments have equal \(n\)th derivative at the junction points.
  • Tangents have equal direction and magnitude at the junction points.
Continuity Description
\(C^-1\) Curve is discontinuous
\(C^0\) Curve is continuous
\(C^1\) First Derivative is continuous
\(C^2\) Second Derivative is continuous
\(C^n\) \(n\)th Derivative is continuous

Geometric Continuity \(G^n\): Tangents have equal direction but not magnitude at the junction points. Curve is \(G^n\) continuous if it can be reparametrized to be \(C^n\) continuous.

Continuity Description
\(G^0\) Curve touch at the joint point (=\(C^0\))
\(G^1\) Curves share a common tangent direction
\(G^2\) Curves share a common center of curvature