ODE:= -2 sin(1/2 theta(t)) cos(1/2 theta(t)) (diff(theta(t),t)^2-9.8000 sin(theta(t))-(150+4 sin(1/2 theta(t))^2) (diff(theta(t),t,t)=0:

 ICS:=  theta(0) = Pi/6, D(theta)(0) = DthetaZero:

The worksheet below shows a river whose velocity is a function of the distance from its mid point to either bank.

In this worksheet, as an example, a boat of fixed speed crosses the river heading at all times towards its destination which is directly across the river from its starting point. The boat's path is animated.

I presume an experienced boat captain could constantly vary his heading so as to cross in a minimum time.

Can a function for this minimum crossing time heading be found and the worksheet altered to show the minimum time path?

  Rivercrossing.mw

plot3d of procedure Sievert correctly displays the constant curvature Sievert surface, but the procedure uses the deprecated command evalm.

What Maple 2016 statement(s) would create the same value of X in Sievert?

Sievert := proc (B)

local a, b, denom, m, X;

a := sinh(B)*u; b := cosh(B)*v;

denom := sinh(B)*((cosh(2*a)-cos(2*b))*cosh(2*B)+2+cosh(2*a)+cos(2*b));

m := cosh(B)*[sinh(a), sin(b)*cos(v), sin(b)*sin(v)]+[0, -cos(b)*sin(v), cos(b)*cos(v)];

X := evalm([u, 0, 0]-8*cosh(B)*cosh(a)*m/denom);

end proc:

plot3d(Sievert(.75), u = -2.5 .. 2.5, v = -10.5 .. 10.5, scaling = constrained, grid = [30, 100], style = patch, shading = xy, lightmodel = light3, orientation = [-3, 140], title = "Sievert's surface", titlefont = [Courier, bold, 14]);

What coding will display a spherical cap of an arbitrary sphere (an arbitrarily located centre and arbitrary radius) where the cap covers an arbitrary solid angle and the radius to the cap's centre is any arbitrary radius of the sphere?

Assume the sphere is defined by its radius and its azimuth and polar angles.

The following are the equation and plot3d of the minimal surface named Catalan:

Catalan := [alpha-sin(alpha)*cosh(beta), 1-cos(alpha)*cosh(beta), 4*sin((1/2)*alpha)*sinh((1/2)*beta)]
plot3d(Catalan, alpha = 0 .. 2*Pi, beta = -3 .. 3, scaling = constrained, title = "Catalan's surface", titlefont = [Courier, bold, 14])

I would like to color and thicken the edge of this surface to emphasize what I presume is the wire which frames the soap film which forms the surface.

Contourplot3d only shows contours at constant values of the z coordinate, however the plot3d display with style option "surface with line" clearly shows the surface's edge contour, among others.

Given the surface equation, is there any way to display the surface edge programmatically e.g. as a spacecurve?