By Adrian Bowyer
Programming for special effects calls for various uncomplicated geometric operations. the obvious method to application those is usually inefficient or numerical volatile. This ebook describes the simplest ways to those user-friendly approaches, delivering the programmer with geometric strategies in a kind that may be without delay integrated into this system being written. it's without delay appropriate to special effects, but in addition to different programming initiatives the place geometric operations are required
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Additional info for A Programmer's Geometry
Cosine functions must be used, at some expense in processor The drawback is that t h e sine and t i m e , to calculate the endpoint coordinates. * y = X j + r c o s ( 6 K) K K = y j ' + sin( V As an alternative, t h e tangent of t h e half-angle, t a n ( Ö / 2 ) , may be stored for each end of t h e arc. This corresponds to t h e uneven but computationally efficient circle parameterisation mentioned in Section 2 1 . To maintain t h e parameter t a n ( 9 / 2 ) in t h e range 0 to 1 , only one quadrant of t h e circle can be represented.
0)/TANT ELSE 56 Τ = 2 . -ACCY) TANT = THEN -XKJ/YKJ Τ = 3 . 0 + Τ = 3 . T2) THEN Neareat point ia on the Neareat point ia an arc ELSE endpoint ENDIF ENDIF If t h e nearest point is on t h e arc t h e n t h e distance from t h e point to t h e arc is simply. r = VC(x K - ν" +( y K " y ) j 2] " Γ ϋ A negative distance indicates that J is within t h e circle. 4 to find t h e χ and y coordinates corresponding to T1 and Τ2. The minimum of these two distances is taken as t h e answer. 2. a r c / s e g m e n t pair is quick This operation If t h e infinite line and t h e whole circle do not intersect, rejection of t h e If, however, t h e line s e g m e n t and arc a r e both short compared to t h e circle radius, many cases w h e r e t h e segment and t h e arc a r e quite distant from each other will remain υ η rejected until quite late in t h e computation.
T h e s a m e is t r u e for If t h e arc occupies several quadrants t h e coordinates of t h e e x t r e m e χ and y values of t h e circle become t h e χ and y coordinates of o n e or m o r e edges of t h e box. For example: T h e following table shows which values must be used to extend t h e box for given start and finish quadrants of t h e arc. 59 Arc begins t t t = = 0-1 = 0-1 t XMIN YMIN XMIN XMAX YMAX XMAX 1-2 YMAX t = 2-3 = XMIN 1-2 = 3-4 YMIN 2-3 XMIN YMIN XMAX YMAX XMIN are to be subjected to repeated boxing - 3-4 XMAX YMIN XMAX YMAX XMAX XMIN YMIN XMIN XMAX YMAX XMAX YMAX YMIN XMIN YMIN XMAX YMAX YMIN YMAX If arcs t YMIN XMAX XMIN XMIN = YMIN YMAX t t tests, t h e n storing their YMAX end points will save computing time.
A Programmer's Geometry by Adrian Bowyer