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Curvature: Cartesian version

x f x 0.5 x * cos * 23 x 2^* 1 + ln * + :

x y x f :

Point P

x.P 5 :

y.P x.P y : 4.6557

Derivatives

x y' x y x diff :

x y'' x y' x diff :

Tangent

m x.P y' :

x tg m x x.P - (* y.P + :

x c x y'' 1 x y' 2^+ (3^sqrt / :

Signed curvature

Center of Osculating Circle

x.C y.C 21 mat x.P y.P 21 mat 1 x.P c 1 x.P y' 2^+ sqrt * / x.P y' - 121 mat * + : 24.8064 17.7913 21 mat

Osculating Circle

TO 02 π * (0.1 range :

XO YO 12 mat TO cos x.P c abs / x.C + vectorize TO sin x.P c abs / y.C + vectorize 12 mat eval :

x f x tg XO YO augment x.C y.C x.P y.P 2 2 mat x.P y.P o 10 red augment x.C y.C o 10 green augment 6 1 sys x c

Curvature: Parametric version

t x t : t y t f : 21 sys

The easiest way to parametrize y=f(x)

T 030 0.1 range :

k 1 T rows range :

You can use uppercase for vectors. Even k it's one, it's more than a counter.

With index

X k el T k el x eval : Y k el T k el y eval : 21 sys

M X Y augment :

M M rows row 30 6.9879 - 12 mat

With vectorize

X T x vectorize eval : Y T y vectorize eval : 21 sys

M X Y augment :

M M rows row 30 6.9879 - 12 mat

dy dx / dy dt / dt dx / * ≡

Point P

t.P 5 :

x.P y.P 21 mat t.P x t.P y 21 mat :

y dy dx / x x.P - (* y.P + ≡

Derivatives

t x' t x t diff :

t y' t y t diff :

t x'' t x' t diff :

t y'' t y' t diff :

Tangent

m t.P y' t.P x' / :

tg k el m X k el x.P - (* y.P + eval :

tg m T x x.P - (* y.P + vectorize eval :

x tg m x x.P - (* y.P + :

and plot it as tg(x)

t c t x' t y'' * t y' t x'' * - t x' 2^t y' 2^+ (3^sqrt / :

The curvature have sign.

x.C y.C 21 mat x.P y.P 21 mat 1 t.P c t.P x' t.P y' 21 mat norme * / t.P y' - t.P x' 21 mat * + : 24.8064 17.7913 21 mat

Center of Osculating Circle

TO 02 π * (0.1 range :

XO YO 12 mat TO cos t.P c abs / x.C + vectorize TO sin t.P c abs / y.C + vectorize 12 mat eval :

M x tg XO YO augment x.C y.C x.P y.P 2 2 mat x.P y.P o 10 red augment x.C y.C o 10 green augment 6 1 sys x T 1 el > ( x T T rows el < ( & x c not defined if

Alvaro

4 appVersion 1.2.9018.0