ECS 178 -- Geometric Modeling Unit 2 -- The Bézier Curve Have Fun with this Curve. It is cool!!
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# Unit 2 -- The Bézier Curve

Section Description Video Notes Exercises

## Preliminaries

Tutorial Notes on Bernstein Polynomials.
Here are some general properties of Bernstein Polynomials, in notes form.

Parametric Curves
The usual y=f(x) methods for curve generation are not general enough for modeling purposes, so we tend to rely on parametric curve methods.

A Divide and Conquer Geometric Approach to Generate a Curve: Using three control points, we describe a divide-and-conquer approach to generate simple curves. This curve is commonly known as a Bézier Curve.
Generating Points on the Curve Directly: We examine the basic step of the divide-and-conquer approach, and generalize this step so that we can generate points on our curve directly.
The Equation of the Bézier Curve: Using this new generalized step, we figure out the analytic form of this curve. It turns out that the curve is just a parabola.
The Geometric Definition of the Bézier Curve: We write down the geometric formula for this curve. The geometric formula can be stated as a recursive method, and formulas similar to this one will be used in the discussion on B-splines.
Properties of the Bézier Curve: Properties of this simple Bézier Curve. Several properties of the curve are easy to see, and we derive them here.

## The Cubic Bézier Curve

A Divide and Conquer Geometric Approach to Generate a Curve: We can use the divide-and-conquer approach to generate more complex curves. We just have to do more steps to get points on the curve.
Generating Points on the Curve Directly: We can generalize this again, and end up with a simple procedure to generate points on a curve. This gives us immediately both a geometric and an analytic formulation of the curve.
Properties of the cubic Bézier curve: We can examine properties of the cubic curve in the same way as we did with the quadratic curve, they are just slightly more general.
A Matrix form of the Bézier Curve: We can write down an interesting matrix form of the curve. We will use this formulation a lot when we deal with subdivision curves and surfaces.
Interpolating Bézier Curves: Only two control points are guaranteed to be on the Bezier curve -- the first and last control points. What if we have a set of points and we want the curve to interpolate these points. We can use the matrix formulation to generate the control points of an interpolating curve.

## The General Bézier Curve

The Analytic and Geometric Forms of the General Bézier Curve: Writing down the geometric and analytic formula for the Bézier curve defined by an arbitrary number of control points.
Why we need more than the Bézier Curve to produce good models: What is good, and not good about the Bézier Curve. Why do we need more than this curve?

## OpenGL and Bézier Curves

Incorporating Bézier curves into your graphics programs: What functions do we use from OpenGL to draw Bézier Curves?