By Adrian Bowyer

Programming for special effects calls for loads of basic geometric operations. the obvious strategy to application those is frequently inefficient or numerical volatile. This publication describes the easiest ways to those straight forward tactics, supplying the programmer with geometric recommendations in a kind that may be without delay integrated into this system being written. it's without delay acceptable to special effects, but additionally to different programming initiatives the place geometric operations are required

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2 6 Tangents b e t w e e n Two Circles If t h e tangent b e t w e e n t h e circles is ax + by + c = 0 t h e n its coefficients a r e given by. a = ( T L r * Λ Γ " V ( y • L · y K > * ( \ · V \ "V b - ( T L r * Λ -V + ( X L · , Τ κ Γ- axK - + (+ \ ( y L - " V V 2 + • (yL - ( - ± r L * Φ 2 V*«\ - " Ζ \ - V (xL - χ / ο 2 2 2 - ( ± L r * Φ y / byK 33 T h e signs attached to t h e two circle radii d e t e r m i n e which of t h e four possible tangents is to be found. If t h e two circles intersect, only two tangents a r e possible: Attempting to calculate either of t h e non-existant tangents will lead to a negative expression to be square rooted in t h e formula.

1. 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.

The second is useful for holding t h e intersections of a ray with t h e objects in a t h r e e - d i m e n s i o n a l scene. 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.