By Graham J. O. Jameson
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Example text
9-4) is a special case of an epicycloid. Fig. 19. Parametric equations: ⎧ ⎛ a − b⎞ φ ⎪x = (a − b) cos φ + b cos ⎜ ⎝ b ⎟⎠ ⎪ ⎨ ⎛ a − b⎞ ⎪ ⎪y = (a − b) sin φ − b sin ⎜⎝ b ⎟⎠ φ ⎩ This is the curve described by a point P on a circle of radius b as it rolls on the inside of a circle of radius a. If b = a/4, the curve is that of Fig. 9-3. Fig. 20. Parametric equations: { x = aφ − b sin φ y = a − b cos φ This is the curve described by a point P at distance b from the center of a circle of radius a as the circle rolls on the x axis.
17. ⎧x = l1x ′ + l2 y′ + l3 z ′ + x 0 ⎪ ⎨y = m1x ′ + m2 y′ + m3 z ′ + y0 ⎪⎩z = n1x ′ + n2 y′ + n3 z ′ + z 0 z ⎧x ′ = l1 ( x − x 0 ) + m1 ( y − y0 ) + n1 (z − z0 ) ⎪ or ⎨y′ = l2 ( x − x 0 ) + m2 ( y − y0 ) + n2 (z − z 0 ) ⎪⎩z ′ = l3 ( x − x 0 ) + m3 ( y − y0 ) + n3 (z − z 0 ) y' O' (x0 , y0 , z0) y O x' where the origin O′ of the x′y′z′ system has coordinates (x0, y0, z0) relative to the xyz system and l1 , m1 , n1; l2 , m2 , n2 ; l3 , m3 , n3 z' x Fig. 10-6 are the direction cosines of the x′, y′, z′ axes relative to the x, y, z axes, respectively.
R = ⑀D p = 1 − ⑀ cos θ 1 − ⑀ cos θ The conic is (i) an ellipse if ⑀ < 1 (ii) a parabola if ⑀ = 1 (iii) a hyperbola if ⑀ > 1 Fig. 18. 19. 20. 21. Eccentricity = ⑀ = c a2 − b2 = a a Fig. 22. 23. Equation in polar coordinates if C is at O: r 2 = a2b2 a sin θ + b 2 cos 2 θ 2 2 8-24. 25. If P is any point on the ellipse, PF + PF′ = 2a If the major axis is parallel to the y axis, interchange x and y in the above or replace u by 12 π − θ (or 90° − u). 26. (y − y0)2 = 4a(x − x0) if parabola opens to right (Fig.