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

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

The Quadratic Bézier Curve

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. Video Here Notes Here Exercise1
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. Video Here Notes Here  
Equation of the Curve 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. Video Here Notes Here  
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. Video Here Notes Here  
Properties of the curve 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. Video Here Notes 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. Video Here Notes Here  
Generating a point 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. Video Here Notes Here Exercise1
Convex Hull 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. Video Here Notes Here  
Matrix Form 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. Video Here Notes Here  
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. Video Here Notes Here  

The General Bézier Curve

Arbitrary 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. Video Here Notes Here  
We need more than the Bezier Curve 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? Video Here    

OpenGL and Bézier Curves

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

 
 

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