Dr. Dobb's Journal - August 2008 - (Page 32) State of the Art ELLIPSE SPECIFICATION USING VECTORS continued from page 30 piecewise, each piece being determined by four control points—the two preceding and two following the curve segment in sequence. Figure 3 shows a sample spline curve segment along with its control points. The interpolating TSpline used in this algorithm is defined by a set of parametric equations (which form a basis curve) and control points, each arranged in vertical/horizontal vectors that are combined by means of matrix multiplication. They are arranged in vectors and given by the equations in Figure 4. The TSpline basis is formed from a single trigonometric function with a period equal to four units. Each unit segment is then shifted to [0,1] to form the four basis spline function curves (Figure 5). Note in the upper graph that all of the zero crossings occur on unit intervals, except for the central part of the graph, which has a value of 1.0. The result can be seen in the lower graph where F2 and F3 have values of 1.0, which when multiplied by P2 and P3 yield the coordinates of P2 and P3 exactly, ensuring that the resulting curve segment interpolates P2 and P3. (Multiply the x and y components of P2 and P3 by 1.0, and you will get P2 and P3, respectively. The other curve segments, multiplied by 0.0, will contribute exactly 0.0 to the calculation.) Rendering Ellipses Figure 1: “Eyeballing” placement of the nearest corner of the bounding box. Figure 2: Approximating with a vector-based ellipse. A typical uniform spline is applied over a set of control points P1, P2, …, Pi, …, Pn–1, Pn D ow nl oa d P2 P3 P1 P4 PEG® GUI Development Tools A family of portable graphics software for designing a high performance GUI for any embedded device. a 1.0 Figure 3: Sample spline curve segment along with its control points. F1 0.0 –0.125 –2.0 F2 –1.0 F3 0.0 F3 F2 1.0 F4 2.0 1.0 application development 0.0 –0.125 1.0 F1 F4 G R APHI C S S O F T WAR E FO R E MBE D D E D SYS TE MS Figure 5: Four basis spline function curves. 32 Dr. Dobb’s Journal l www.ddj.com l August 2008 http://www.swellsoftware.com 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 Dr. Dobb's Journal - August 2008 - Dr. Dobb's Journal - August 2008 (Page Cover1) Dr. Dobb's Journal - August 2008 - Dr. Dobb's Journal - August 2008 (Page Cover2) Dr. Dobb's Journal - August 2008 - Dr. Dobb's Journal - August 2008 (Page 1) Dr. Dobb's Journal - August 2008 - Dr. Dobb's Journal - August 2008 (Page 2) Dr. Dobb's Journal - August 2008 - Dr. Dobb's Journal - August 2008 (Page 3) Dr. Dobb's Journal - August 2008 - Contents (Page 4) Dr. Dobb's Journal - August 2008 - Contents (Page 5) Dr. Dobb's Journal - August 2008 - Friday Night Fish Fry (Page 6) Dr. Dobb's Journal - August 2008 - Friday Night Fish Fry (Page 7) Dr. Dobb's Journal - August 2008 - Friday Night Fish Fry (Page 8) Dr. Dobb's Journal - August 2008 - Friday Night Fish Fry (Page 9) Dr. Dobb's Journal - August 2008 - Alia Vox (Page 10) Dr. Dobb's Journal - August 2008 - Alia Vox (Page 11) Dr. Dobb's Journal - August 2008 - Developer Diaries (Page 12) Dr. Dobb's Journal - August 2008 - Developer Diaries (Page 13) Dr. Dobb's Journal - August 2008 - Developer’s Notebook (Page 14) Dr. Dobb's Journal - August 2008 - Developer’s Notebook (Page 15) Dr. Dobb's Journal - August 2008 - A Conversation with Christos Papadimitriou (Page 16) Dr. Dobb's Journal - August 2008 - A Conversation with Christos Papadimitriou (Page 17) Dr. Dobb's Journal - August 2008 - A Conversation with Christos Papadimitriou (Page 18) Dr. Dobb's Journal - August 2008 - A Conversation with Christos Papadimitriou (Page 19) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 20) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 21) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 22) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 23) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 24) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 25) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 26) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 27) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 28) Dr. Dobb's Journal - August 2008 - OpenGL and Mobile Devices: Round 2 (Page 29) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 30) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 31) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 32) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 33) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 34) Dr. Dobb's Journal - August 2008 - Ellipse Specification Using Vectors (Page 35) Dr. Dobb's Journal - August 2008 - Embed Custom GUIs in WPF (Page 36) Dr. Dobb's Journal - August 2008 - Embed Custom GUIs in WPF (Page 37) Dr. Dobb's Journal - August 2008 - Embed Custom GUIs in WPF (Page 38) Dr. Dobb's Journal - August 2008 - Embed Custom GUIs in WPF (Page 39) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 40) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 41) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 42) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 43) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 44) Dr. Dobb's Journal - August 2008 - Building RIAs on J2EE Foundations (Page 45) Dr. Dobb's Journal - August 2008 - Disentangling Concepts in Object-Oriented Systems (Page 46) Dr. Dobb's Journal - August 2008 - Disentangling Concepts in Object-Oriented Systems (Page 47) Dr. Dobb's Journal - August 2008 - Disentangling Concepts in Object-Oriented Systems (Page 48) Dr. Dobb's Journal - August 2008 - Disentangling Concepts in Object-Oriented Systems (Page 49) Dr. Dobb's Journal - August 2008 - The Agile Edge (Page 50) Dr. Dobb's Journal - August 2008 - The Agile Edge (Page 51) Dr. Dobb's Journal - August 2008 - The Agile Edge (Page 52) Dr. Dobb's Journal - August 2008 - Effective Concurrency (Page 53) Dr. Dobb's Journal - August 2008 - Effective Concurrency (Page 54) Dr. Dobb's Journal - August 2008 - Effective Concurrency (Page 55) Dr. Dobb's Journal - August 2008 - Swaine’s Flames (Page 56) Dr. Dobb's Journal - August 2008 - Swaine’s Flames (Page Cover3) Dr. Dobb's Journal - August 2008 - Swaine’s Flames (Page Cover4)
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