Rotation of Ellipsoid

Transcription

1 15 Rotation An object may be rotated in space. Any rotation has a direction which is always assumed to be ccw. Any rotation also has an axis which is always assumed to be aligned with a spatial dimension (x,y,z). A single rotation is defined by the axis of rotation (x, or y, or z) and the angle of rotation (φ, or θ, or ψ). A spatial point (x,y,z) transforms to the point (X,Y,Z). The rotation of an ellipse (2D) and an ellipsoid (3D) are discussed. Single Rotation; A matrix (A xφ, or A yθ, or A zψ ) is associated with each single rotation. A null rotation is also represented as a matrix; A 0 Rotation around the x axis is; A xφ = x y 0 Cosφ -Sinφ z 0 Sinφ Cosφ X = x Y = ycosφ + zsinφ Z = -ysinφ + zcosφ Rotation around the y axis is; A yθ = x Cosθ 0 Sinθ y z -Sinθ 0 Cosθ X = xcosθ - zsinθ Y = y Z = xsinθ + zcosθ December 23, 2018 Page 1

2 Rotation around the z axis is; A zψ = x Cosψ -Sinψ 0 y Sinψ Cosψ 0 z The null rotation is; A 0 = x y z X = x Y = y Multiple Rotation; A triple rotation (A 3 ) is represented as the product of three single rotations (a rotation around each spatial axis); A 3 = A zψ A yθ A xφ A 3 = CosθCosψ -CosφSinψ + SinφSinθCosψ SinφSinψ + CosφSinθCosψ CosθSinψ CosφCosψ + SinφSinθSinψ -SinφCosψ + CosφSinθSinψ -Sinθ SinφCosθ CosφCosθ The transforms are; X = x CosθCosψ + ycosθsinψ - zsinθ Y = xsinφsinθcosψ - xcosφsinψ + ycosφcosψ + ysinφsinθsinψ + zsinφcosθ Z = xsinφsinψ + xcosφsinθcosψ + ycosφsinθsinψ - ysinφcosψ + zcosφcosθ December 23, 2018 Page 2

3 A double rotation (A 2 ) is represented by setting one angle to zero (assume θ = 0). Giving; Sinθ = 0 and Cosθ = 1 A 2 = x Cosψ -CosφSinψ SinφSinψ y Sinψ CosφCosψ -SinφCosψ z 0 Sinφ Cosφ The transforms are; Y = -xcosφsinψ + ycosφcosψ + zsinφ Z = xsinφsinψ - ysinφcosψ + zcosφ The Standard Ellipse; A standard ellipse is; x 2 /a 2 + y 2 /b 2 = 1 Where; x,y are dimensional variables a,b are properties of the ellipse a is the major radius b is the minor radius If; x=0 Then; y = ±b If; y=0 Then; x = ±a Focus; x = ±(a 2 b 2 ) ½, y = 0 The Rotated Ellipse; A rotation around the z axis gives the 2D transform; = 0 The rotated ellipse is; X 2 /a 2 + Y 2 /b 2 = 1 (xcosψ + ysinψ) 2 /a 2 + (ycosψ - xsinψ) 2 /b 2 = 1 Translation (h,k) gives; [(x-h)cosψ + (y-k)sinψ] 2 /a 2 + [(y-k)cosψ (x-h)sinψ] 2 /b 2 = 1 December 23, 2018 Page 3

4 The Standard Ellipsoid; A standard ellipsoid is; x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1 Where; x,y,z are dimensions of space a,b,c are properties of the ellipse Focus; x = ±(a 2 b 2 ) ½, y = 0 It is convenient to define an ellipsoid with a circular mid-section; c = b Single ; A single rotation around the z axis gives the 3D transforms; The rotated ellipsoid is; X 2 /a 2 + Y 2 /b 2 + Z 2 /c 2 = 1 Assume c = b; (xcosψ + ysinψ) 2 /a 2 + (ycosψ - xsinψ) 2 /b 2 + z 2 /b 2 = 1 Translation (h,k,l) gives; [(x-h)cosψ + (y-k)sinψ] 2 /a 2 + [(y-k)cosψ (x-h)sinψ] 2 /b 2 + (z-l) 2 /b 2 = 1 Double ; A double rotation is assumed to be a combination of two single rotations. A single rotation around the z axis may be combined with and a single rotation around the x axis. Giving a 3D transform; Y = -xcosφsinψ + ycosφcosψ + zsinφ Z = xsinφsinψ - ysinφcosψ + zcosφ The rotated ellipsoid is; X 2 /a 2 + Y 2 /b 2 + Z 2 /c 2 = 1 Assume c = b; [xcosψ + ysinψ] 2 /a 2 + [ycosφcosψ - xcosφsinψ + zsinφ] 2 /b 2 + [xsinφsinψ - ysinφcosψ + zcosφ] 2 /b 2 = 1 Displacement (h,k,l) gives; [(x-h)cosψ + (y-k)sinψ] 2 /a 2 + [(y-k)cosφcosψ - (x-h)cosφsinψ + (z-l)sinφ] 2 /b 2 + [(x-h)sinφsinψ - (y-k)sinφcosψ + (z-l)cosφ] 2 /b 2 = 1 December 23, 2018 Page 4

5 Midsection Circle; If; X = 0 and c = b Then the midsection of the ellipsoid is defined as a circle; Y 2 + Z 2 = b 2 Giving the equation of a 2D circle in 3D space; (ycosφcosψ - xcosφsinψ + zsinφ) 2 + (xsinφsinψ - ysinφcosψ + zcosφ) 2 = b 2 Expanding terms; b 2 = x 2 Cos 2 φsin 2 ψ + y 2 Cos 2 φcos 2 ψ + z 2 Sin 2 φ + 2xzSinφSinψCosφ - 2xySin 2 φsinψcosψ - 2yzSinφCosψCosφ + x 2 Sin 2 φsin 2 ψ + y 2 Sin 2 φcos 2 ψ + z 2 Cos 2 φ + 2yzSinφCosψCosφ - 2xySin 2 φsinψcosψ - 2xzSinφSinψCosφ Recombine; b 2 = x 2 Sin 2 ψ + y 2 Cos 2 ψ + z 2-4xySin 2 φsinψcosψ b 2 = (xsinψ - ycosψ) 2 + z 2-4xySin 2 φsinψcosψ + 2xySinψCosψ b 2 = (xsinψ - ycosψ) 2 + z 2 + 2xySinψCosψ(1-2Sin 2 φ) Giving the equation of a 2D circle in 3D space; b 2 = (xsinψ - ycosψ) 2 + z 2 + 2xySinψCosψ(Cos 2 φ - Sin 2 φ) Conclusion; A rotational transformation and a translational displacement will manipulate the orientation and position of an ellipsoid in 3D space. The equation of the midsection of the ellipsoid will give the equation of a 2D circle in 3D space. December 23, 2018 Page 5