A Programmer's Geometry by Adrian Bowyer

By Adrian Bowyer

Programming for special effects calls for a good number of basic geometric operations. the obvious method to software those is frequently inefficient or numerical volatile. This publication describes the simplest techniques to those effortless strategies, offering the programmer with geometric options in a kind that may be without delay integrated into this system being written. it really is at once appropriate to special effects, but in addition to different programming initiatives the place geometric operations are required.

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T h e value of t at t h e intersection points is fix t = J O J 2 2 - χ ) • g(y - y j ± V { r | V • g ) - [ « y - y ) - gfc - χ ) ] } 1 J 2 2 (f • g ) 0 O J O J and t h e points a r e found by substituting t h e s e values back into t h e parametric equations. -ACCY) THEN ..... 0, and t h e code can be simplified. 2 2 3 Intersections of Two Circles Two circles may have two intersection points or one intersection point at a common t a n g e n t Alternatively they may not intersect at all. T h e position of t h e intersection points may be found by applying Pythagoras' t h e o r e m , which will give t h e parametric equation of t h e line on which the intersection points lie, and t h e n solving t h e resulting quadratic equation in t h e p a r a m e t e r as was done in t h e last section.

The two signs from t h e root give t h e two cases on o n e side of t h e line. 43 A z e r o root indicates that only one tangent circle is possible in t h e specified region. 2), w e have code of t h e form: CDASH = C + A*XJ + B*YJ IF ( CDASH. LT. GE. ACCY) THEN X = XJ + ATEMP*CFAC Only one circle Y — YJ + BTEMP*CFAC ELSE ROOT = SQRT( ROOT ) XCONST - XJ - ATEMP*CFAC YCONST = YJ - BTEMP*CFAC 44 Two solutions possible XVAR = BTEMP *ROOT YVAR = ATEMP *ROOT XI = XCONST - XVAR Yl = YCONST + YVAR X 2 - XCONST + XVAR Y 2 = YCONST - YVAR ENDIF ENDIF 2 1 2 Circles of Given Radius Tangent to Two Circles This problem reduces to that given in Section 2 1 0 by transforming t h e given radii to make one of t h e fixed circies into a point with zero radius: r' = r + r I I J r · = 0 J r · = r - r Κ Κ J If this is done the centre of the new circle of radius centre of r.

2 Distance from a Point to a Line Segment Κ The distance from a point to a line s e g m e n t is t h e distance to t h e line only if t h e normal from t h e point to t h e line strikes t h e line b e t w e e n t h e s e g m e n t endpoints. Otherwise it is t h e distance from t h e point to t h e nearest s e g m e n t e n d p o i n t It is convenient to consider t h e line as t h e interval t - 0 to t=1 of an infinite parametric line: χ = χ y = y Κ K + t(x L - χ ) Κ • t(yL - y If w e then calculate t h e value of t w h e r e t h e normal from t h e point J strikes t h e line, values between 0 and 1 indicate that t h e closest point is on t h e s e g m e n t , values below 0 indicate that Κ is t h e closest point, and values above 1 indicate that L is t h e closest point T h e value of t is found from [(χ t = - κ X - A (χ - V L - χ ) Κ 2 (y * K + (y L - y j K y - y ) Κ L - V 3 2 and if the nearest point is in the s e g m e n t , the distance to it is: 47 r = V([(xK - V • \ + - - V + % · V ] 2} By truncating t h e value of t to b e b e t w e e n 0 and 1 w e can use this formula in all t h r e e cases, coded as follows: XKJ = XK - XJ YKJ = YK - YJ XLK = XL - XK YLK = YL - YK DENOM = X L K * X L K + I F ( DENOM.

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