Bezier Curves

Overview

Bezier curves are parametrically defined by a set of control points (p₀ through p_n) that determine the shape of the curve. You can take advantage of this fact by deriving a Bezier class from the VariableShape class, as you did for Polygons in the basic Lab 4.
 

The (x,y) coordinates of any point on a Bezier curve are determined by evaluation of the parametric equations for the curve (which will be covered in lecture). From the starting point of the curve to the endpoint, the parameter u varies from 0 to 1. For u=0, the coordinates of the curve are p₀, and for u=1, the coordinates of the curve are p_n. These are the only points on the curve that coincide with any control points; the remaining interior of the curve does not pass through any of the remaining control points p₁ through p_n-1. For examples, see the sample curves in the image to the right.

Implementation

The Bezier class should consequently be very similar to your Polygon class; the main difference should be in the Bezier ::Draw(...) function, where you render the curve. You will render (i.e. draw) the curve by using the methods provided in a partially-implemented Bezier class, which you can download here.  These methods implement the parametric equations for Bezier curves of various order. (Note that a QPointArray data format is required by the predefined methods of the Bézier class) You must implement the rest of the class, including the virtual Draw() method.

The getPoint() method will be the workhorse of the Draw() method. To render the curve, your Draw() method must repeatedly call the getPoint() method in order to evaluate the coordinates of points along the curve. Your Draw() method must then draw a series of straight-line segments (using Qt's line drawing methods) between these points, created a faceted approximation (an alias; actually a polyline) to the theoretical curve. As you compute more points, the faceted polyline will more closely approximate the true curve.

Detail Requirements:

• Add a Bezier Curve Mode button (example above) to your GUI that puts your application in curve mode.
• Derive the Bezier class from VariableShape; use the behavior encapsulated in VariableShape to manage the control point data.
  • Similar to your polygon class from last week, much of the varshape code will can be used as-is, without overriding. The main difference from your polygon class will be in the bezier::Draw(...) function.
  • The number of facets used to render the curve mjst be a Bezier constructor parameter that defaults to 100.
    • Optimized functions for Bézier curves with 3 or 4 control point are provided, along with a general function that works for 2 or more control points. You must use the optimized versions when you have only 3 or 4 control points.
    • A Bézier curve with 2 control points degenerates into a line, while a Bézier curve with 1 control point degenerates into a point. These cases should be handled optimally by your Draw function.
• Implement a mouse interface for drawing Bézier curves similar to your Polygon drawing interface, where the curve definition is terminated with a mouse double-click..
• It is NOT REQUIRED that you implement a command line interface for drawing Bézier curves.
• The curve must exhibit the attributes (such as color) that the Bezier class inherits from VariableShape.
• Your Draw method should also draw a polyline between the control points that define the Bezier curve, so that your output is similar to that shown above. The polylines should be drawn as the control points are being entered, but the curve should not be displayed until the entry of control points is terminated (with the double-click).

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