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Shape Parametrization Using Fourier Transforms

Shape data

Reference: Samir Khan,
https://www.maplesoft.com/applications/view.aspx?SID=154546

X 558533512509519538527501485493499499465406413441430406397393392397403406397365356354326311336355315278278249242232204208209204206242242263274287334361320314344355360379406407399393391394404415439431402438487503497486488514538537511508518546 stack :

Y 420412431480522505444431480591605548444416426478516533488395305356463525418414488465416458497495463409488514458512441441485412508493463516437403503482495429416499439407490533392309333452520529499452418416492593606544448429482518508450424414 stack :

n X length : 80

rn 1 n range :

m 0.5 n * Ceil 1 - :

k 1 m range :

Fourier Transform

z FFT Z rn el 1 n / sqrt z h el e 2 i * π * rn 1 - (* h 1 - (* n / -^* h 1 n sum * eval : 11 line :

FX X FFT 11 m range + el :

FY Y FFT 11 m range + el :

Polar forms

rx φx 12 mat FX abs vectorize FX arg Re vectorize 12 mat eval :

ry φy 12 mat FY abs vectorize FY arg Re vectorize 12 mat eval :

Fourier-parametric equations

r φ t finterp r h el π 2 / 2 π * h t * n / * - φ h el + sin * h 1 m sum :

t xf rx φx t finterp :

t yf ry φy t finterp :

Plot

T 0 m 1 + (m 1 + 300 / range : k 1 T length range : Xo k el T k el xf : Yo k el T k el yf : 41 line

Xo Yo augment X Y . augment 2 1 sys

Alvaro