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From: Nmacgre on 13 Apr 2010 22:37
Hello,

Im trying to write an animation in Maple that shows a parabola rlling
on a line, and its focus sketching out a catenary. I was wondering
what the equation was to model such a thing. Any help appreciated.
From: Robert Israel on 14 Apr 2010 01:58
Nmacgre <natemacgregor(a)gmail.com> writes:

> Hello,
>
> Im trying to write an animation in Maple that shows a parabola rlling
> on a line, and its focus sketching out a catenary. I was wondering
> what the equation was to model such a thing. Any help appreciated.

To get you started: let's say the line is x = 0, and the original parabola is
y = x^2. Let L(t) be the arc length on that parabola from (0,0) to the point
(t, t^2). That point on the parabola must get moved to (L(t), 0), and the
parabola rotated to the right around that point by angle arctan(2*t).
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Nmacgre on 15 Apr 2010 14:19
On Apr 13, 10:58 pm, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> Nmacgre <natemacgre...(a)gmail.com> writes:
> > Hello,
>
> > Im trying to write an animation in Maple that shows a parabola rlling
> > on a line, and its focus sketching out a catenary. I was wondering
> > what the equation was to model such a thing. Any help appreciated.
>
> To get you started: let's say the line is x = 0, and the original parabola is
> y = x^2.  Let L(t) be the arc length on that parabola from (0,0) to the point
> (t, t^2).  That point on the parabola must get moved to (L(t), 0), and the
> parabola rotated to the right around that point by angle arctan(2*t).  
> --
> Robert Israel              isr...(a)math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

Thanks for the help. I'm assuming you meant the line y=0? I did some
research based on what you said, and it looks like I'm going to need
some parametric equations for x and y, and the use of some rotation
matricies? Any more help would be appreciated. I don't have a degree
in math, so thanks for your patience.

From: Robert Israel on 15 Apr 2010 15:13

> On Apr 13, 10:58=A0pm, Robert Israel
> <isr...(a)math.MyUniversitysInitials.ca> wrote:
> > Nmacgre <natemacgre...(a)gmail.com> writes:
> > > Hello,
> >
> > > Im trying to write an animation in Maple that shows a parabola rlling
> > > on a line, and its focus sketching out a catenary. I was wondering
> > > what the equation was to model such a thing. Any help appreciated.
> >
> > To get you started: let's say the line is x =3D 0, and the original
> > parab=
> ola is
> > y =3D x^2. =A0Let L(t) be the arc length on that parabola from (0,0) to
> > t=
> he point
> > (t, t^2). =A0That point on the parabola must get moved to (L(t), 0), and
> > =
> the
> > parabola rotated to the right around that point by angle arctan(2*t). =A0
> > --
> > Robert Israel =A0 =A0 =A0 =A0 =A0 =A0
> > =A0isr...(a)math.MyUniversitysInitial=
> s.ca
> > Department of Mathematics =A0 =A0 =A0 =A0http://www.math.ubc.ca/~israel
> > University of British Columbia =A0 =A0 =A0 =A0 =A0 =A0Vancouver, BC,
> > Cana=
> da
>
> Thanks for the help. I'm assuming you meant the line y=3D0? I did some
> research based on what you said, and it looks like I'm going to need
> some parametric equations for x and y, and the use of some rotation
> matricies? Any more help would be appreciated. I don't have a degree
> in math, so thanks for your patience.
>

Sorry, yes, I meant y=0.
Actually you can do all the translating and rotating using the plottools
package. Thus:

> with(plots): with(plottools):
L:= unapply(int(sqrt(1+(2*x)^2),x=0..X),X);
P1:= display([pointplot([0,1/4]), plot(x^2,x=-2..2)]):
F:= t -> rotate(translate(P1,L(t)-t,-t^2),-arctan(2*t),[L(t),0]);
animate(F,[t],t=-2..2,scaling=constrained);
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Nmacgre on 17 Apr 2010 16:11
On Apr 15, 12:13 pm, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> > On Apr 13, 10:58=A0pm, Robert Israel
> > <isr...(a)math.MyUniversitysInitials.ca> wrote:
> > > Nmacgre <natemacgre...(a)gmail.com> writes:
> > > > Hello,
>
> > > > Im trying to write an animation in Maple that shows a parabola rlling
> > > > on a line, and its focus sketching out a catenary. I was wondering
> > > > what the equation was to model such a thing. Any help appreciated.
>
> > > To get you started: let's say the line is x =3D 0, and the original
> > > parab=
> > ola is
> > > y =3D x^2. =A0Let L(t) be the arc length on that parabola from (0,0) to
> > > t=
> > he point
> > > (t, t^2). =A0That point on the parabola must get moved to (L(t), 0), and
> > > =
> > the
> > > parabola rotated to the right around that point by angle arctan(2*t). =A0
> > > --
> > > Robert Israel =A0 =A0 =A0 =A0 =A0 =A0
> > > =A0isr...(a)math.MyUniversitysInitial=
> > s.ca
> > > Department of Mathematics =A0 =A0 =A0 =A0http://www.math.ubc.ca/~israel
> > > University of British Columbia =A0 =A0 =A0 =A0 =A0 =A0Vancouver, BC,
> > > Cana=
> > da
>
> > Thanks for the help. I'm assuming you meant the line y=3D0? I did some
> > research based on what you said, and it looks like I'm going to need
> > some parametric equations for x and y, and the use of some rotation
> > matricies? Any more help would be appreciated. I don't have a degree
> > in math, so thanks for your patience.
>
> Sorry, yes, I meant y=0.  
> Actually you can do all the translating and rotating using the plottools
> package.  Thus:
>
> > with(plots): with(plottools):
>
>   L:= unapply(int(sqrt(1+(2*x)^2),x=0..X),X);
>   P1:= display([pointplot([0,1/4]), plot(x^2,x=-2..2)]):
>   F:= t -> rotate(translate(P1,L(t)-t,-t^2),-arctan(2*t),[L(t),0]);
>   animate(F,[t],t=-2..2,scaling=constrained);
> --
> Robert Israel              isr...(a)math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada- Hide quoted text -
>
> - Show quoted text -

Thanks Robert, works like a champ.
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