By Adrian Bowyer
Programming for special effects calls for various easy geometric operations. the obvious solution to software those is usually inefficient or numerical volatile. This ebook describes the easiest methods to those hassle-free tactics, offering the programmer with geometric recommendations in a sort that may be at once included into this system being written. it really is without delay acceptable to special effects, but in addition to different programming projects the place geometric operations are required
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Extra resources for A Programmer's Geometry
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.
24 Tangents from a Point to a Circle If t h e point is outside t h e circle t h e r e a r e two tangents to it, if it is just on t h e circumference t h e r e is one, and if it is inside t h e r e is none. If t h e equation of t h e tangent line required is ax + by + c = 0 Then t h e coefficients a and b a r e obtained from: (y ( - ν - K - ν * *κ - ν w (χ T r ( y y } b = K K " j * Κ - χ ) 2 J " 2 + ( y K + (y κ - y ) (χ V 2 + (y K - 2] J (y κ - Γ /- κ 2 κ - V *<»K - * / * K (χ y y y / - r 2 ] K / and c can then be calculated from t h e fact that t h e tangent passes through J: 29 c = -ax J - by J T h e r e a r e normally two possible tangent lines, obtained by attaching a sign to r .
An obvious alternative is to store the arc's c e n t r e , radius, and two angles corresponding to the end points. So long as a conventional direction is established (preferably anticlockwise) this allows arcs through any angle with no danger of inconsistency or ambiguity. 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.