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Notation: - a vector : $[a,b,c,...]$ - 1-norm of a vector: $ ||[x,y,z]||_1 = \sqrt[1] {|x^1|+|y^1|+|z^1|} $ or $ |x|+|y|+|z| $ - or for : $ [v_1,v_2,v_3,..] $, the 1-norm is: $ \sum_{i=1}^n |v_i| $ - 2-norm of a vector: $ ||[x,y,z]||_2 = \sqrt { x^2+y^2+z^2 } $ - or for: $ [v_1,v_2,v_3,..] $, the 2-norm is: $ \sqrt { \sum_{i=1}^n v_i^2 } $ - k-norm of a vector: $ ||[x,y,z]||_k = \sqrt[k] { x^k+y^k+z^k } $ - or for: $ [v_1,v_2,v_3,..] $, the k-norm is: $ \sqrt[k] { \sum_{i=1}^n |v_i^k| } $ --- ## axis-angle to AAA given axis-angle: $ A=[ {x},{y},{z},{{\theta}}] $, $A_r=[x,y,z]$, and $A_\theta = \theta$. Angle angle angle: $ \frac {{A_\theta}A_r} {||A_r||_1} $ --- ## AAA to axis-angle $ A_a = [{X},{Y} ,{Z}] $ Axis-Angle: $ [ \frac {A_a} {||A_a||_2}, || {A_a} ||_1 ] $ --- ## This is A(1-dT) + B(dT) ### A given: $ A=[ {x_1},{y_1},{z_1},{{\theta}_2}] $ Convert to AAA : $ A_{a}= \frac {{{\theta}_1} [x_1,y_1,z_1]} {|| [x_1,y_1,z_1] ||_1} $ - Expanded angle-angle-angle result: $ [ \frac { {{\theta}_1} x_1 } {|x_1|+|y_1|+|z_1|} , \frac { {{\theta}_1} y_1 } {|x_1|+|y_1|+|z_1|} , \frac { {{\theta}_1} z_1 } {|x_1|+|y_1|+|z_1|} ] $ ### B given: $ B=[ {x_2},{y_2},{z_2},{{\theta}_2}] $ Convert to AAA : $ B_{a}= \frac {{{\theta}_2}[x_2,y_2,z_2]} { || [x_2,y_2,z_2] ||_1 } $ - Expanded angle-angle-angle result: $ [ \frac { {{\theta}_2} x_2 } {|x_2|+|y_2|+|z_2|} , \frac { {{\theta}_2} y_2 } {|x_2|+|y_2|+|z_2|} , \frac { {{\theta}_2} z_2 } {|x_2|+|y_2|+|z_2|} ] $ ### Result C Add vectors... with dT: (P=VT+D); or Position equals velocity(B) times time(dT) plus a base Distance (A). $ C_{a} = A_{a} d_T + B_{a} ({1-d_T}) = [{X_3},{Y_3} ,{Z_3}] $ - or: $ [{x_1 d_T +x_2 ({1-d_T})},{y_1 d_T +y_2 ({1-d_T})} ,{z_1 d_T +z_2 ({1-d_T})}] $ - Expanded: $[\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|} , \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|}, \frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|} ] $ result: $ C = [ \frac {X_3} {||C_a||_2}, \frac {Y_3} {||C_a||_2}, \frac {Z_3} {||C_a||_2}, {||{C_a}||_1} ] $ --- - Expanded angle: $ |\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 d_T } {|x_2|+|y_2|+|z_2|} | +| \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 d_T } {|x_2|+|y_2|+|z_2|}| +|\frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 d_T} {|x_2|+|y_2|+|z_2|} | $ - Expanded axis: $ \frac {\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 d_T } {|x_2|+|y_2|+|z_2|} } {\sqrt { (\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 d_T } {|x_2|+|y_2|+|z_2|} )^2 + ( \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 d_T } {|x_2|+|y_2|+|z_2|})^2 + \frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 d_T} {|x_2|+|y_2|+|z_2|})^2 } }, \frac {\frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 d_T } {|x_2|+|y_2|+|z_2|}} {\sqrt { (\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 d_T } {|x_2|+|y_2|+|z_2|} )^2 + ( \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 d_T } {|x_2|+|y_2|+|z_2|})^2 + \frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 d_T} {|x_2|+|y_2|+|z_2|})^2 }} , \frac {\frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 d_T} {|x_2|+|y_2|+|z_2|}} {\sqrt { (\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 d_T } {|x_2|+|y_2|+|z_2|} )^2 + ( \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 d_T } {|x_2|+|y_2|+|z_2|})^2 + \frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 d_T} {|x_2|+|y_2|+|z_2|})^2 } } $ --- (Working space, to get expaneded expressions above... ) - $||C_a||_1$ Expanded: $ |\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|} | +| \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|}| +|\frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|} | $ computation of the 2-norm to scale the axis: - $||C_a||_2$ Expanded: $\sqrt { (\frac { {{\theta}_1} x_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} x_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|} )^2 + ( \frac { {{\theta}_1} y_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} y_2 ({1-d_T}) } {|x_2|+|y_2|+|z_2|})^2 + \frac { {{\theta}_1} z_1 d_T} {|x_1|+|y_1|+|z_1|} + \frac { {{\theta}_2} z_2 ({1-d_T})} {|x_2|+|y_2|+|z_2|})^2 } $
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Sun, 23 Aug 2020 12:46 GMT