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

0761f068-2caa-4fc7-bf01-009051d4f983 12 Alvaro Diaz Falconi 4 5 false false 0 0 decimal <content> <p>Homogeneous curve</p> </content>

Homogeneous curve degree

d 15 :

d 2 + d C 1 - d 2 + (! d! (d 2 + d - (! * / 1 - ≡

\0020\ d 1 + (d 2 + (* 2 / 1 - ≡

Independent term

α 1 :

Number of points

n 0.5 d 1 + (* d 2 + (* 1 - : 135

X n Random : 0.5003 0.5399 0.9721 0.8658 0.6125 51 mat

Y n Random : 0.9635 0.8406 0.5955 0.8746 0.5787 51 mat

Random poings

Homogeneous exponents

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 000000000000000123456789101112131415215 mat

v rows 135

The condition \[REGION[de277c9392814712b94ed04913548360]]\ excludes (0,0), which represents α

r c + 0 d ltle

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^y 9^y 10^y 11^y 12^y 13^y 14^y 15^115 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 0.9635 0.9284 0.8946 0.862 0.8306 0.8406 0.7066 0.5939 0.4992 0.4196 0.5955 0.3546 0.2111 0.1257 0.0749 0.8746 0.7649 0.669 0.5851 0.5117 0.5787 0.3349 0.1939 0.1122 0.0649 0.9073 0.8231 0.7468 0.6775 0.6147 0.001 1.0534 106 -^* 1.0812 109 -^* 1.11 1012 -^* 1.139 1015 -^* 0.0778 0.0061 0.0005 3.6699 105 -^* 2.8564 106 -^* 0.2656 0.0706 0.0187 0.005 0.0013 0.7603 0.5781 0.4395 0.3342 0.2541 0.4138 0.1712 0.0709 0.0293 0.0121 0.267 0.0713 0.019 0.0051 0.0014 125 mat

Solution

C M n 1 matrix α - dn_LinAlgSolve :

C transpose 8.5554 - 29.3148 - 702.5728 4026.5328 - 11541.8379 15 mat

C M 1 -^n 1 matrix α - (* :

enable this only for n small.

C transpose 12.0856 - 43.9231 52.1404 - 11.0626 5.7659 15 mat

Curve equation

x y f C x y φ transpose * α + :

d 15

n 135

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

Alvaro

4 appVersion 1.0.8348.30405