Dr. Dobb's Journal - August 2008 - (Page 30)

State of the Art by Robert F. Kauffmann Ellipse Specification Using Vectors Robert is a software engineer and award-winning independent animator. He can be reached at mathart63@gmail.com. An alternative method for rendering ellipses Over the years, I’ve made heavy use of graphical editing programs and sometimes needed to use ellipse drawing functions. What I’ve found is that ellipse drawing functions in graphical editing programs typically use bounding boxes for specifying ellipses. This method serves well if the task requires fitting the ellipse within the confines of a box, or if it’s not necessary to align the edge of the ellipse with precision. On occasion, however, I’ve had a need to use an ellipse (or a portion thereof) to approximate a freeform curve. In such cases, bounding boxes serve poorly. Why? Because the outer corner of the bounding box that specifies the ellipse is far enough away from the curve of interest that “eyeballing” placement of the nearest corner of the bounding box to specify an ellipse that approximates the freeform curve is error prone and usually produces erroneous results, as in Figure 1. The problem is that ellipse curves specified by bounding boxes are physically disconnected from the opposite corners of the box that are actually under user control. This physical disconnect robs users of the ability to place the edge of the ellipse precisely against the freeform curve of interest. To give users the control needed to produce ellipses that approximate the curvature of a freeform curve, it is more expedient to specify an ellipse using points that interpolate (lie on the actual ellipse) rather than exterpolate (in which the ellipse is tied to the points but detached from them). This lets users click directly on the curve and anchor the ellipse to the freeform curve of interest from the outset. One approach is to specify an ellipse from a vector the user specifies. In Figure 2, the user clicks on some point lying upon the freeform curve and drags the end of the vector—along with the corresponding ellipse—until it closely approximates the freeform curve. You can implement this scheme using a uniform interpolating trigonometric spline (TSpline) whose control points lie at the corners of a rectangle. Alternatively, you can use the corners of a rhombus (see “Implementing Uniform Trigonometric Spline Curves”; www.ddj.com/architect/184410198). Both approaches inscribe (rather than circumscribe) the ellipse. In either case, a single vector is sufficient to define the four control points that specify an ellipse formed of a TSpline. For this algorithm, the most practical choice is to implement the TSpline ellipse that is specified by an inscribed rectangle. Trigonometric Splines For the purpose of the application I present here, think of splines as a means used to mathematically “connect the dots” with a smooth curve in some application where you are interested in defining a curve of arbitrary shape using movable points to control the shape of the curve. An interpolating uniform spline such as the Catmull-Rom spline, or the interpolating TSpline used in this algorithm, is controlled by a set of control points (commonly known as “knots”). Splines are defined 30 Dr. Dobb’s Journal l www.ddj.com l August 2008 http://www.ddj.com/architect/184410198 http://www.ddj.com

Table of Contents Feed for the Digital Edition of Dr. Dobb's Journal - August 2008

Dr. Dobb's Journal - August 2008 Contents Friday Night Fish Fry Alia Vox Developer Diaries Developer’s Notebook A Conversation with Christos Papadimitriou OpenGL and Mobile Devices: Round 2 Ellipse Specification Using Vectors Embed Custom GUIs in WPF Building RIAs on J2EE Foundations Disentangling Concepts in Object-Oriented Systems The Agile Edge Effective Concurrency Swaine’s Flames

Dr. Dobb's Journal - August 2008

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