kill(all)$

van=facebook.com/LearningCalculus$

      Analysis: Best Shape Solar Concentrator for Solar Smelter
      L. Van Warrren
      Friday, June 15, 2018

      Equation 1: General Form of Parabola with focus at x = h and y = p + k

equation1: (x - h)^2 = 4*p * (y - k);

      General Equation of Parabolic Dish or Trough Profile

f(x,p,h,k):=''(rhs(first(solve(equation1,y))));

      Without loss of generality we can pin the vertex
      of the parabola at the origin (x=0,y=0)

y(x,p):=f(x,p, 0,0)$
y(x,p);

      Local slope of a point on the surface of the parabola at a station x

yp(x,p):=''(diff(y(x,p),x));

      Local surface normal at a point on the surface of the Parabola

n(x,p):= -1/yp(x,p)$ n(x,p);

      Arc Length of a Parabolic Segment gives us Cost
      For a dish of width a and focal point p we compute
      the length of material to form a trough.

      Assume that parabola is of finite size,
      Integrate arc length to compute segment length

assume(a>0,p>0)$
arcLength(a,p):=''(fullratsimp(integrate(sqrt(1+(x/(2*p))^2),x,0,a)));

      The inverse hyperbolic function is needed for plotting results:

arsinh(u):=log(u + sqrt(u^2 + 1))$

      This is the formula for the arc length of a parabola of half-width x and focus p

arclen(x,p):=substitute(arsinh(x/(2*p)),asinh(x/(2*p)),arcLength(x,p))$
arclen(x,p);

      The cost is proportioanal to the surface area of the materials.
      We add a term of p itself to account for the structure that holds the target.
      It turns out that the minimum cost parabolic trough is always one whose
      width equals its focal length.

cost(a,p):=arclen(a,p)+p$
cost(a,p);

cost(0,1);

float(cost(0.1,1));

f(p,p,0,0);

fp(p,p);

180/%pi*atan(2);

float(%);

180/%pi*atan(3/4);

float(%);