adjusting manipulate output in mathematica - wolfram-mathematica

Question

I need help. I have many variables, that I use in my Graphics[] command, that are dependent of one variable (H in my example). I want to manipulate my graphic so that by changing value of H graphic changes accordingly. But it is not as easy as I've thought.
If you have any idea on how to acomplish this, I would be grateful.
(*This variables are dependent on H that I want to change in
manipulate*)
R = 10;
\[Alpha] = ArcSin[H/R];
p = H/Tan[\[Alpha]];
n = 1.5;
\[Beta] = ArcSin[n Sin[\[Alpha]]];
\[Theta] = \[Beta] - \[Alpha];
l = H/Tan[\[Theta]];
(*This is the graphic I want to make manipulated*)
Graphics[{(*Incident ray*)Line[{{-2, H}, {p, H}}],(*Prism*)
Circle[{0, 0}, R, {0, Pi/2}],
Line[{{0, 0}, {0, 10}}],(*Refracted ray*)
Line[{{p, H}, {p + l, 0}}],(*Surface*)
Line[{{0, 0}, {p + l + 10, 0}}]}]
Here's one of my solutions but it's really messy. What I did is just manually pluged in those values. Is there any more appropriate way to acomplish this:
R = 10;
n = 1.5;
Manipulate[
Graphics[{(*Incident ray*)
Line[{{-2, H}, {H/Tan[ArcSin[H/10]], H}}],(*Prism*)
Circle[{0, 0}, R, {0, Pi/2}],
Line[{{0, 0}, {0, 10}}],(*Refracted ray*)
Line[{{H/Tan[ArcSin[H/10]],
H}, {H/Tan[ArcSin[H/10]] +
H/Tan[ArcSin[n Sin[ArcSin[H/10]]] - ArcSin[H/10]],
0}}],(*Surface*)
Line[{{0,
0}, {H/Tan[ArcSin[H/10]] +
H/Tan[ArcSin[n Sin[ArcSin[H/10]]] - ArcSin[H/10]] + 10,
0}}]}], {H, 0.0001, 10, Appearance -> "Labeled"}]
And also how to make my graphic not to change it's size constantly. I want prism to have fixed size and incident ray to change its position (as it happens when H gets > 6.66 in my example above / this solution).
The question is maybe confusing, but if you try it in Mathematica, you'll see what I want. Thank you for any suggestions.

Answer 1

I think your solution is not bad in general, Mark already noticed in his reply. I loved simplicity of Mark's solution too. Just for the sake of experiment I share my ideas too.
1) It is always a good thing to localize your variables for a specific Manipulate, so their values do not leak and interfere with other dynamic content. It matters if you have additional computation in your notebook - they may start resetting each other.
2) In this particular case if you try to get read of extra variables plugging expressions one into each other your equations became complicated and it is hard to see why they would fail some times. A bit of algebra with help of functions TrigExpand and FullSimplify may help to clarify that your variable H has limitations depending on refraction index value n (see below).
3) If we are aware of point (2) we can make variable n dynamic too and link the value H to n (resetting upper bound of H) right in the controls definition, so always it should be H<10/n . If[..] is also necessary so the controls will not “pink”.
4) If your formulas would depend on R we could also make R dynamic. But I do not have this information, so I localized R via concept of a “dummy“ control (ControlType -> None) – which is quite useful concept for Manipulate.
5) Use PlotRange and ImageSize to stop jiggling of graphics
6) Make it beautiful ;-)
These points would be important if you’d like, for example, to submit a Demonstration to the Wolfram Demonstration Project. If you are just playing around – I think yours and Mark’s solutions are very good.
Thanks,
Vitaliy
Manipulate[If[H >= 10/n, H = 10/n - .0001]; Graphics[{
{Red, Thick, Line[{{-2, H}, {Sqrt[100 - H^2], H}}]},
{Blue, Opacity[.5], Disk[{0, 0}, R, {0, Pi/2}]},
{Red, Thick, Line[{{Sqrt[100 - H^2], H},
{(100 n)/(Sqrt[100 - H^2] n - Sqrt[100 - H^2 n^2]), 0}}]}},
Axes -> True, PlotRange -> {{0, 30}, {0, 10}},
ImageSize -> {600, 200}], {{R, 10}, ControlType -> None},
{{n, 1.5, "Refraction"}, 1.001, 2, Appearance -> "Labeled"},
{{H, 3, "Length"}, 0.0001, 10/n - .0001, Appearance -> "Labeled"}]

Answer 2

I think your first batch of code looks fine and is easy to place into a Manipulate. I would recommend use of the PlotRange option in Graphics.
R = 10;
n = 1.5;
Manipulate[
\[Alpha] = ArcSin[H/R];
p = H/Tan[\[Alpha]];
\[Beta] = ArcSin[n Sin[\[Alpha]]];
\[Theta] = \[Beta] - \[Alpha];
l = H/Tan[\[Theta]];
Graphics[{
Line[{{-2, H}, {p, H}}],(*Prism*)
Circle[{0, 0}, R, {0, Pi/2}],
Line[{{0, 0}, {0, 10}}],(*Refracted ray*)
Line[{{p, H}, {p + l, 0}}],(*Surface*)
Line[{{0, 0}, {p + l + 10, 0}}]},
PlotRange -> {{-1,33},{-1,11}}],
{H,0.0001,6,Appearance->"Labeled"}]