AutoCAD's Spline and understanding Bézier Curves: From Ellipses to Hyperbolas

Many engineers and designers have used Spline Fit or Spline CV commands without ever wondering about the mathematics behind them. To make sense of this fascinating topic, let’s take a look at how Bézier curves are defined and how they can be applied to create conic sections such as ellipses, parabolas, and hyperbolas.

The Mathematics of Rational Bézier Curves

Rational Bézier curves can be described by the following equation:

We are interested in the equation when n = 2, and then it looks like this:

In the above equation, we used the following notations:

P_i = (x_i, y_i) - control points

w_i - weight parameter

t₀ - curve parameter, ranging from 0 to 1

Linking Theory with Software

When working with splines in CAD software, you saw a panel of curve properties:

Let’s now analyze what each piece of data in the panel represents in relation to the formula.

• Number of points corresponds to the number of terms in the numerator and denominator of the Bézier equation.

• Control point X, Y, Z values represent the coordinates of P_i. While the formula indexes points from 0 to 2, the software may label them from 1 to 3.

• Weight corresponds to the w_i parameter of each control point.

By adjusting these parameters, you can generate different conic curves:

• If w₀ = w₂ = 1 and 0 < w₁ > 0.5 the curve is an ellipse.

• If w₀ = w₁ = w₂ = 1 the curve is an parabola

• If w₁ > 0.5 the curve is an hyperbola

Constructing a Hyperbola with Bézier Curves

Let’s examine the hyperbola in the figure below.

In the figure, we can see two focal points of hyperbola, and also semi-major axis a = 64 and semi-minor axis b = 48.

To model this hyperbola using Spline CV, we must determine the positions of the three control points.

The equation of the hyperbola is:

The equation of the tangent to the hyperbola is:

Let us look at some of Spline properties:

An important property of the spline is that the lines P₁P₀ and P₂P₁ are tangents at points P₀ and P₂. If we apply the tangent equation at point P₀, we obtain the following expression:

Because P₁ = (x₁, 0)

If the control points have coordinates (x₀, y₀), (x₁, 0), (x₀, -y₀) and the weight parameters are 1, w₁, 1, then, based on the rational Bézier curve, the value of the vertex can be written as:

After simplification, the above equation becomes:

Now, we have 2 equations

If we apply the equation of the hyperbola at the point P₀ = (x₀, y₀), we obtain the following equation:

When solving the system of three equations with three unknowns, we obtain the coordinates for the P₀point.

These equations are valid exclusively for the positions of the control points forming an equilateral triangle and for values of w₁ > 1.

This is the final shape of the hyperbola when the calculated coordinates for w1 = 2 are applied.

Why This Matters

Understanding the connection between mathematical equations and CAD tools not only helps engineers design more precisely but also deepens appreciation for the hidden elegance of mathematics. Bézier curves are used everywhere, from car design to computer graphics. They are more than just smooth lines. They are a bridge between abstract equations and real-world shapes.

If you wish to learn more, you can watch this video by Freya Holmér The Continuity of Splines. And of course, see our YouTube series AutoCAD in Math

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