<content> <p>Homogeneous curve</p> </content>

0761f068-2caa-4fc7-bf01-009051d4f983 5 Alvaro Diaz Falconi 4 5 false false 0 0 decimal <content> <p>Homogeneous curve</p> </content>

Homogeneous curve degree

d 8 :

Independent term

α 1 :

Number of points

n 0.5 d 1 + (* d 2 + (* 1 - : 44

X n Random : 0.4582 0.622 0.2608 0.8378 0.503 51 mat

Y n Random : 0.4173 0.9791 0.4817 0.5349 0.3395 51 mat

Random poings

Homogeneous coefficients

v 02 matrix eval : r 0 d range c 0 d range r c + 0 d ltle v v r c 12 mat stack : if for for 21 line

v transpose 000000001111111123456780123456215 mat

Homogeneous curve

x y φ V 0 : k 1 n range V 1 k el x v k 1 el^y v k 2 el^* : for V 31 line :

x y φ y y 2^y 3^y 4^y 5^y 6^y 7^y 8^x x y * x y 2^* x y 3^* x y 4^* x y 5^* 114 mat

Coefficients matrix

M 0 n matrix eval : k 1 n range M M X k el Y k el φ stack eval : for 21 line

M 0.4173 0.1741 0.0727 0.0303 0.0127 0.9791 0.9586 0.9385 0.9189 0.8996 0.4817 0.232 0.1118 0.0538 0.0259 0.5349 0.2861 0.1531 0.0819 0.0438 0.3395 0.1153 0.0391 0.0133 0.0045 0.5421 0.2939 0.1593 0.0864 0.0468 0.1621 0.0263 0.0043 0.0007 0.0001 0.1377 0.019 0.0026 0.0004 4.9531 105 -^* 0.3546 0.1257 0.0446 0.0158 0.0056 0.8226 0.6767 0.5567 0.458 0.3768 0.6874 0.4726 0.3249 0.2233 0.1535 0.0078 6.0372 105 -^* 4.6908 107 -^* 3.6447 109 -^* 2.8319 1011 -^* 125 mat

Solution

C M n 1 matrix α - dn_LinAlgSolve :

C transpose 28.6525 - 312.1959 1596.9473 - 4431.2423 7102.4411 - 15 mat

C M 1 -^n 1 matrix α - (* :

disable this only if n > 8

C transpose 28.6523 - 312.1932 1596.9298 - 4431.1877 7102.3474 - 15 mat

Curve equation

x y f C x y φ transpose * α + :

X Y . 4 red augment x y f 2 1 sys

Alvaro

4 appVersion 1.0.8348.30405