A Programmer's Geometry by Adrian Bowyer

By Adrian Bowyer

Programming for special effects calls for lots of basic geometric operations. the obvious method to application those is usually inefficient or numerical volatile. This e-book describes the simplest techniques to those straight forward techniques, delivering the programmer with geometric concepts in a sort that may be without delay integrated into this system being written. it really is without delay acceptable to special effects, but in addition to different programming initiatives the place geometric operations are required

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The second r^ t e r m s affect t h e centre position similarly with respect to t h e second line. O/DETERM X - (B2*C1RAB2 - B1*C2RAB1)*DETINV Y = (A1*C2RAB2 - A2*C1RAB1) *DETINV END IF 2 1 0 Circles of Given Radius through a Point and Tangent to a Circle One of the given points (circle c e n t r e , K, or J) is m a d e a local origin for the calculation; w e chosen the point J. • x KJ The circle c e n t r e coordinates may then be found from [(x KJ • y 2(x y = y. • y KJ [(x KJ 2 KJ • y 2(x 40 ) - r KJ KJ 2 KJ • y (2r I /J • r η K ± y • r ηj K' ^ χ lKJ 2 ) KJ ) - r • y Κ (2r κ ι 2 KJ ) IKJ y have and The sign of is i m p o r t a n t diagram above.

1 Representation of a Line S e g m e n t The representation of straight line segments is achieved most simply by storing each endpoints. values of t h e parameter, t, which bound t h e s e g m e n t segments of t h e same line a r e required: 44 segment's An alternative is to store t h e parametric equation of t h e infinite straight line and t h e This is of particular value w h e r e a number of It allows t h e intersections to be g e n e r a t e d in any order and then sorted, segments being then described by consecutive pairs of values.

OR. 0) THEN Intersection not within ELSE X = XK + XLX*S Y = YK + YLK*S 50 line segments s>1, t>1 ENDIF ENDIF If this procedure is being used to find t h e intersections of many line segments with o n e single line segment then two of t h e differences ( e g XLK and YLK) need not be recomputed for each comparison. If an appreciable number of t h e lines being compared a r e either horizontal or vertical, t h e n it will be worth branching to much simpler comparisons with such lines. For instance, if t h e line M N is vertical t h e n t h e intersection point is given by χ = χ M y = y Κ + (χ M - χ Ky K L - y ) Κ \ -V which saves seven multiplications and a division over t h e g e n e r a l method.

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