1, x->1, by -> 1,bx->0.8}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=1.8;
$dime="WIDTH=600 HEIGHT=600";
$texte="Simple Spiral
The below logarithmic spiral has the polar equation:
r = a.eθ cot α
You can move the red and yellow control points to respectively vary
the spiral's characteristic angle α and magnification a.
Notice that when moving the yellow point, the spiral does not change
in shape, only in scaling."; break;
case "Pedal":
$Mfile="pedal.m";
$indep_var = "{ax -> 0.0001,x->0.0001, by -> 1,bx->0.8}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=1;
$dime="WIDTH=600 HEIGHT=600";
$texte="Pedal
The pedal is given by the intersection
between the tangent line to the original spiral with the perpendicular
to this line which goes on the pole. The results of the calculations
give this equation (With r = a.eθ cot α
the equation of the original spiral):
rp =
sin α . a.e(θ-dθ) cot α
with the phase
difference : dθ = α - ½ π Move the
control points to see how the pedal move with the basic spiral. ";
break;
case "Roulette":
$Mfile="roulette.m";
$indep_var = "{ax -> 0.00001, x->0.00001, by -> 1,bx->1.}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=1;
$dime="WIDTH=600 HEIGHT=600";
$texte="Roulette
The roulette is given by the intersection between the tangent line to
the original spiral with perpendicular line to the radius of the
original spiral which goes on the pole. The results of the
calculations give this equation (With r = a.eθ cot
α the equation of the original spiral):
rr = tan α a.e(θ-dθ) cot α
with the phase difference : dθ = - ½ . π
Move the control points to see how the roulette move with the basic spiral.
";
break;
case "Inverse":
$Mfile="inverse.m";
$indep_var = "{ax -> 1, x->1, by -> 1,bx->1.1}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=1;
$dime="WIDTH=600 HEIGHT=600";
$texte="Spiral inverse
The inverse is the spiral obtain with the same scaling factor but with
the opposite characteristic angle. The results of the calculations
give this equation (With r = a.eθ cot α
the equation of the original spiral):
ri =
a.eθ cot -α
with the phase difference :
dθ = 0 Move the control points to see how the
inverse move with the basic spiral.
";
break;
case "Catacaustic":
$Mfile="catacaustic.m";
$indep_var = "{ax ->5, x->5, by -> 30,bx->10}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=1;
$dime="WIDTH=600 HEIGHT=600";
$texte="Catacaustic
The catacaustic of a curve C with respect to a point O
is obtained as follows: at O we place a light source (here,
O is the pole of the spiral, i.e. the black point). Each ray
(here, the yellow lines) must travel a distance d to reach
C, and thereafter it will be reflected. The catacaustic is the
locus of the points obtained by moving the distance d along the
reflected rays.
Assuming that C has the polar equation r =
a.eθ cot α, the catacaustic has polar
equation
rc = 2.cos α a.e(θ--dθ) cot α
with the phase difference : dθ = α
and you can notice that this is also a logarithmic spiral with same
characteristic angle.
";
break;
case "Evolute":
$Mfile="evolute.m";
$indep_var = "{ax -> 1, x->1, by -> 1,bx->0.5}";
$dep_var = "{coef->(bx-x)/by, bx->(coef*by)+ax, x -> ax}";
$zoom=2;
$dime="WIDTH=600 HEIGHT=600";
$texte="Spiral evolutes
The interior evolute of a curve C is the locus of centers of
circles of curvature of C. The exterior evolute is the
symmetric to the interior evolute with respect to C. Assuming
that C has the polar eqation r = a.eθ cot
α, these evolutes respectively have the equations:
rint = cot α a.e(θ-dθint) cot
α
with the phase difference : dθint = π / 2
rext = √(cot2 α + 4)
a.e(θ-dθext) cot α
with the phase difference : dθext = - arcsin(cot α / √(cot2
α + 4))
As depicted on the applet, both evolutes have same characteristic
angle as the original spiral.
";
break;
}
?>
Logarithmic spiral derivatives
Go to applet
list
Go to applet
list