<p>Plot</p>

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Point Orthogonal Projection onto a flat or spatial Curve

The cases are considered when the given point is very far from the curve

Calculation procedure 1. We will use Draghilev's method a) to parameterize the curves and find a good initial guess. We will use the point a0, found by the minimum distance from the curve to a given point P. b) to calculate the unit vector of the tangent T (x) to a point on the curve. 2 To the system of equations we add an equation that expresses the conditions of perpendicularity of the tangent at the point x of the curve and the segment connecting the points P and x. This means that the inner product of the unit tangent vector to the curve f (x) and the unit vector Px is equal to zero. As a result, we obtain the system of equations \[REGION[0]]\ 3. We solve the system with the initial assumption a0

x F x f x T Px Px norme / * 21 mat :

plot2 B P1 1 el P1 2 el a1 1 el a1 2 el 22 mat a4 1 el a4 2 el . 5 black 15 mat P4 1 el P4 2 el a4 1 el a4 2 el 22 mat a1 1 el a1 2 el . 5 black 15 mat P3 1 el P3 2 el a3 1 el a3 2 el 22 mat a3 1 el a3 2 el . 5 black 15 mat a2 1 el a2 2 el . 5 black 15 mat P2 1 el P2 2 el a2 1 el a2 2 el 22 mat 101 sys :

plot3 B γ * 1 col B γ * 2 col augment Pr4 γ * 1 col Pr4 γ * 2 col P4 γ * 1 col P4 γ * 2 col 22 mat Pr2 γ * 1 col 1 el Pr2 γ * 2 col 1 el . 4 red 15 mat Pr3 γ * 1 col 1 el Pr3 γ * 2 col 1 el . 4 red 15 mat Pr4 γ * 1 col 1 el Pr4 γ * 2 col 1 el . 4 red 15 mat Pr1 γ * 1 col Pr1 γ * 2 col P1 γ * 1 col P1 γ * 2 col 22 mat Pr3 γ * 1 col Pr3 γ * 2 col P3 γ * 1 col P3 γ * 2 col 22 mat Pr1 γ * 1 col 1 el Pr1 γ * 2 col 1 el . 4 red 15 mat Pr2 γ * 1 col Pr2 γ * 2 col P2 γ * 1 col P2 γ * 2 col 22 mat Pr1 γ * 1 col 1 el Pr1 γ * 2 col 1 el 110 black 15 mat Pr2 γ * 1 col 1 el Pr2 γ * 2 col 1 el 210 black 15 mat Pr3 γ * 1 col 1 el Pr3 γ * 2 col 1 el 310 black 15 mat Pr4 γ * 1 col 1 el Pr4 γ * 2 col 1 el 410 black 15 mat 131 sys :

γ 0.4208 0.5237 - 0.454 0.3673 0.2668 - 0.891 - 0.5878 0.809 033 mat :

x0 tmax Δt P D f x f : n f rows 1 + : k 1 n range : u k el x k el : N tmax Δt / : c 1 n range : 16 mat x J f u Jacobian : b 1 - (x J 1 n 1 - n n submatrix * : 12 mat A x J 1 n 1 - 1 n 1 - submatrix : sA A 1 n 1 - k 1 + n 1 - submatrix : X0 x0 x f al_nleqsolve : 12 mat n 2 > V 1 el b A 1 n 1 - 2 n 1 - submatrix augment det - : (V n 1 - el A 1 n 1 - 1 n 2 - submatrix b augment det - : V n el A det - : 12 mat k 2 : k n 2 - ≤ k k 1 + : V k el A 1 n 1 - 1 k 1 - submatrix b sA augment det - : for 31 line V 1 el b 1 el : V 2 el A det : 12 mat if Unit Tangent Vector x T V V norme / : t x v x T : 13 mat B X0 0 tmax N t x v rkfixed 1 N 2 n 1 + submatrix : k 1 N range : dist k el P B k row - norme eval : Px c el P c el x c el - : 13 mat a0 B dist augment n 1 + csort 1 row c el transpose eval : 11 mat Adding\0020\orthogonality\0020\equation x F x f x T Px Px norme / * 11 mat stack : 12 mat Pr a0 transpose x F al_nleqsolve transpose : Pr B Pr F 13 mat 121 line :

P L x0 tmax Δt P D :

Example 1

P1 56594512 mat :

P2 1200 - 600012 mat :

P3 1 - 20 - 12 mat :

P4 3.5 28 - 12 mat :

x f 2 x 1 el (2^* 4 x 1 el * x 2 el * + 4 x 2 el (3^* + 26 x 2 el * - 22 - 18 x 1 el x 2 el * sin * + 11 mat :

x0 105 - 12 mat transpose : Δt 0.1 : tmax 56 : 13 mat

a1 B a1 F 13 mat P1 L :

a1 2.1903 2.5675 12 mat

a1 F transpose 2.1136 108 -^* 5.396 109 -^* 12 mat

a2 B a2 F 13 mat P2 L :

a2 4.3664 - 3.3081 12 mat

a2 F transpose 1.028 108 -^* 3.6598 1010 -^* 12 mat

a3 B a3 F 13 mat P3 L :

a3 5.263 - 3.8824 - 12 mat

a3 F transpose 2.0278 108 -^* - 7.4853 1010 -^* 12 mat

a4 B a4 F 13 mat P4 L :

a4 7.9327 2.9704 - 12 mat

a4 F transpose 5.8563 109 -^* 5.1738 1010 -^* 12 mat

plot2

Example 2

P1 1469412 mat :

P2 16794 - 12 mat :

P3 36.7 172 - 12 mat :

P4 4318612 mat :

x f x 1 el 2 - (2^5 x 2 el (2^* + x 1 el (2^x 2 el (10^* + 37 x 1 el * x 2 el * sin * - 9 - 11 mat :

x0 21 - 12 mat transpose : Δt 0.01 : tmax 30 : 13 mat

a1 B a1 F 13 mat P1 L :

a1 5.3597 0.377 12 mat

a1 F transpose 1.2273 109 -^* 7.1325 108 -^* 12 mat

a2 B a2 F 13 mat P2 L :

a2 5.4024 0.291 - 12 mat

a2 F transpose 4.2184 1010 -^* - 2.365 108 -^* 12 mat

a3 B a3 F 13 mat P3 L :

a3 5.2044 0.4905 - 12 mat

a3 F transpose 2.2694 108 -^* 3.7245 109 -^* - 12 mat

a4 B a4 F 13 mat P4 L :

a4 5.1331 0.5823 12 mat

a4 F transpose 2.2382 108 -^* 4.2928 109 -^* 12 mat

plot2

Example 3

P1 969412 mat :

P2 67 - 94 - 12 mat :

P3 36.7 172 - 12 mat :

P4 43 - 18612 mat :

x f x 1 el (2^x 2 el * x 1 el x 2 el + sin - 11 mat :

x0 5 - 012 mat transpose : Δt 0.1 - : tmax 20 - : 13 mat

a1 B a1 F 13 mat P1 L :

a1 1.2346 0.6283 12 mat

a1 F transpose 7.8431 1010 -^* 4.2628 1011 -^* - 12 mat

a2 B a2 F 13 mat P2 L :

a2 1.4258 - 0.4667 - 12 mat

a2 F transpose 1.0585 108 -^* 3.1513 108 -^* 12 mat

a3 B a3 F 13 mat P3 L :

a3 0.1891 3.2155 - 12 mat

a3 F transpose 1.5088 109 -^* - 5.363 109 -^* - 12 mat

a4 B a4 F 13 mat P4 L :

a4 0.1919 - 3.2148 12 mat

a4 F transpose 3.1089 1010 -^* 2.6101 109 -^* - 12 mat

plot2

Example 4

P1 21727312913 mat :

P2 354376 - 48113 mat :

P3 348 - 677697 - 13 mat :

P4 385 - 347 - 467 - 13 mat :

x f x 1 el (2^x 2 el (2^+ x 3 el (2^+ 9 - x 1 el 9 x 2 el * sin + 2 x 3 el * cos + 21 mat :

x0 12113 mat transpose : Δt 0.05 : tmax 74 : 13 mat

Pr1 B Pr1 F 13 mat P1 L :

Pr1 1.8308 1.9608 1.3429 13 mat

Pr1 F 1.7949 1011 -^* - 1.6162 109 -^* - 1.11 109 -^* 31 mat

Pr2 B Pr2 F 13 mat P2 L :

Pr2 1.4942 1.6229 - 2.0331 13 mat

Pr2 F 3.5147 1010 -^* - 6.0093 109 -^* - 5.1002 1011 -^* - 31 mat

Pr3 B Pr3 F 13 mat P3 L :

Pr3 0.9231 - 1.6137 2.3545 - 13 mat

Pr4 B Pr4 F 13 mat P4 L :

Pr4 1.203 - 1.2273 - 2.459 - 13 mat

plot3

Pr3 F 1.6984 1011 -^* - 2.4526 109 -^* - 8.2835 109 -^* - 31 mat

Pr4 F 1.6984 1011 -^* - 2.4526 109 -^* - 8.2835 109 -^* - 31 mat

Example 5

x f x 1 el (2^4 / x 2 el (2^4 / + sin x 3 el - 0.7097 - x 1 el 2 - (2^x 2 el 2 - (2^+ x 3 el (2^+ 9 - 21 mat :

x0 12113 mat transpose : Δt 0.1 : tmax 74 : 13 mat

Pr1 B Pr1 F 13 mat P1 L :

Pr1 3.7602 4.426 0.1278 13 mat

Pr1 F 6.0736 1011 -^* - 1.7983 1010 -^* 9.956 1011 -^* - 31 mat

Pr2 B Pr2 F 13 mat P2 L :

Pr2 2.7074 0.9097 - 0.1825 13 mat

Pr2 F 1.8386 109 -^* 2.089 109 -^* - 9.0759 1010 -^* - 31 mat

Pr3 B Pr3 F 13 mat P3 L :

Pr3 0.9079 4.212 1.7072 - 13 mat

Pr4 B Pr4 F 13 mat P4 L :

Pr4 0.1159 - 0.006 - 0.7063 - 13 mat

plot3

Pr3 F 1.4826 1010 -^* 6.1922 109 -^* - 1.0655 108 -^* 31 mat

Pr4 F 1.4826 1010 -^* 6.1922 109 -^* - 1.0655 108 -^* 31 mat