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# Equations equation of a line slope intercept explicit

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Unformatted text preview: ines More efficient algorithms exist Solving Simultaneous Equations Equation of a line • Slope-intercept (explicit equation): y = mx + b • Implicit Equation: Ax + By + C = 0 • Parametric Equation: Line defined by two points, P0 and P1 – P(t) = P0 + (P1 - P0) t, where P is a vector [x, y]T (t) t, – x(t) = x0 + (x1 - x0) t – y(t) = x0 + (y1 - y0) t Parametric Line Equation Describes a finite line Works with vertical lines (like the viewport edge) 0 <=t <= 1 • Defines line between P0 and P1 t<0 • Defines line before P0 t>1 • Defines line after P1 Parametric Lines and Clipping Define each line in parametric form: Define • P0(t)…Pn-1(t) Define each edge of viewport in parametric form: • PL(t), PR(t), PT(t), PB(t) Could perform Cohen-Sutherland intersection Could tests using appropriate viewport edge and line tests Line / Edge Clipping Equations Faster line clippers use parametric equations Line 0: • x0 = x00 + (x01 - x00) t0 • y0 = y00 + (y01 - y00) t0 Viewport Edge L: • xL = xL0 + (xL1 - xL0) tL • yL = yL0 + (yL1 - yL0) tL x00 + (x01 - x00) t0 = xL0 + (xL1 - xL0) tL y00 + (y01 - y00) t0 = yL0 + (yL1 - yL0) tL • Solve for t0 and/or tL Cyrus-Beck Algorithm We wish to optimize line/line intersection • Start with parametric equation of line: – P(t) = P0 + (P1 - P0) t • And a point and normal for each edge – PL, NL Cyrus-Beck Algorithm Find t such that PL NL [P(t) - PL] = 0 P(t) P0 Substitute line equation for P(t) Solve for...
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