By Adrian Bowyer
Programming for special effects calls for a great number of easy geometric operations. the obvious strategy to application those is frequently inefficient or numerical risky. This ebook describes the easiest techniques to those simple systems, supplying the programmer with geometric recommendations in a kind that may be at once included into this system being written. it truly is at once acceptable to special effects, but in addition to different programming projects the place geometric operations are required
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The line is normalised. 2. ACCY) Point J lies on the THEN line. ELSE IF ( C D A S H . L T . 0 . -ACCY) THEN Point J is too far from the line. ACCY) THEN X = XJ - ATEMP*CFAC Y = YJ - BTEMP *CFAC One poaaible circle ELSE ROOT = SQRT(ROOT) Two XCONST = XJ - ATEMP *CFAC circles poaaible YCONST = YJ - BTEMP*CFAC XVAR = BTEMP *ROOT YVAR = ATEMP*ROOT XI = XCONST + XVAR Yl = YCONST - YVAR X 2 = XCONST - XVAR Y 2 = YCONST + YVAR ENDIF ENDIF ENDIF 2 9 Circles of Given Radius Tangent to Two Lines As long as t h e lines a r e not parallel (when t h e r e is either no solution, or, if t h e gap b e t w e e n 38 the lines is t h e circle diameter, an infinite number of solutions) t h e n t h e r e a r e four centres for circle of given radius that m a k e it tangential to both t h e lines.
ACCY) THEN . Circles just touch at (X, Y) ELSE ROOT = DSTINV*SQRT(ROOT) Two XFAC = XLK*ROOT intersections YFAC = YLK*ROOT XI = X - YFAC Yl = Y + XFAC X2 = X + YFAC Y2 = Y - XFAC ENDIF ENDIF ENDIF The straight line b e t w e e n t h e points of intersection and, in t h e limit, t h e common tangent is given by t h e implicit equation ax • by + c = 0 where a = χ b 28 L - χ Κ and c = [(r, 2 L - r 2 ) - ( χ Κ L - χ ) 2 - (y Κ L 2 - y ) ] Κ - χ (χ Κ L - χ ) - y (y Κ KL - y ) Κ This equation is not normalised.
If a boxing test fails to eliminate a comparison t h e next step is to calculate t h e intersections of t h e line and t h e whole circle. 6, and t h e n to work out t h e intersections (if any) of t h e resulting parametric line with t h e whole circle, as shown in Section parameters in t h e range 0 to 1 at t h e intersections correspond to intersections inside t h e 22 The corresponding χ and y values a r e used to calculate t h e tangent of t h e angle m a d e by candidate intersection at t h e circle centre.
A Programmer's Geometry by Adrian Bowyer