MATH 223 

Parametrized Surfaces 

 

with(plots); -1 

plot3d([`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), t], s = 0 .. 4, t = 0 .. `+`(`*`(3, `*`(Pi))));  

Plot_2d
 

 

Plot_2d
 

 

f := proc (s, t) options operator, arrow; [`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), t] end proc;  

Typesetting:-mprintslash([f := proc (s, t) options operator, arrow; [`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), t] end proc], [proc (s, t) options operator, arrow; [`*`(s, `*`(cos(t))), `*`(s, `*`(sin(... (1)
 

plot3d(f(s, t), s = 0 .. 4, t = 0 .. `+`(`*`(3, `*`(Pi))));  

Plot_2d
 

 

Parametrization of Tangent Plane

f(s, t);  

[`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), t] (2)
 

f(1, `+`(`*`(`/`(1, 4), `*`(Pi))));  

[`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 4), `*`(Pi)))] (3)
 

fs := proc (s, t) options operator, arrow; [cos(t), sin(t), 0] end proc;  

Typesetting:-mprintslash([fs := proc (s, t) options operator, arrow; [cos(t), sin(t), 0] end proc], [proc (s, t) options operator, arrow; [cos(t), sin(t), 0] end proc]) (4)
 

fs(1, `+`(`*`(`/`(1, 4), `*`(Pi))));  

[`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), 0] (5)
 

ft := proc (s, t) options operator, arrow; [`+`(`-`(`*`(s, `*`(sin(t))))), `*`(s, `*`(cos(t))), 1] end proc;  

Typesetting:-mprintslash([ft := proc (s, t) options operator, arrow; [`+`(`-`(`*`(s, `*`(sin(t))))), `*`(s, `*`(cos(t))), 1] end proc], [proc (s, t) options operator, arrow; [`+`(`-`(`*`(s, `*`(sin(t)... (6)
 

fs(1, `+`(`*`(`/`(1, 4), `*`(Pi))));  

[`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), 0] (7)
 

Tangent Plane 

T := proc (s, t) options operator, arrow; `+`(f(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `*`(fs(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `*`(s)), `*`(ft(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `*`(t))) end proc;  

Typesetting:-mprintslash([T := proc (s, t) options operator, arrow; `+`(f(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `*`(fs(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `*`(s)), `*`(ft(1, `+`(`*`(`/`(1, 4), `*`(Pi)))), `... (8)
 

T(s, t);  

`+`([`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 4), `*`(Pi)))], `*`([`+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`... (9)
 

 

expand(T(s, t));  

[`+`(`-`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2)), `*`(t)))), `*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2)), `*`(s))), `*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2))))), `+`(`*`(`/`(1, 2), `*`(`^`(2, `/`(1, 2)), `*`(t))), `... (10)
 

plot3d(expand(T(s, t)), s = 0 .. 4, t = 0 .. `+`(`*`(3, `*`(Pi))));  

Plot_2d
 

Plot Surface and Tangent Plane near Point of Tangency: 

 

Surface := plot3d(f(s, t), s = 0 .. 2, t = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))), color = blue); -1 

Plane := plot3d(expand(T(s, t)), s = 0 .. 2, t = 0 .. `+`(`*`(`/`(1, 2), `*`(Pi))), color = red); -1 

plots:-display(Surface, Plane);  

Plot_2d
 

 

Expand Domain on t
f(s, t);  

[`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), t] (11)
 

plot3d(f(s, t), s = 0 .. 4, t = 0 .. `+`(`*`(6, `*`(Pi))));  

Plot_2d
 

Replace t with s for 3rd Component 

plot3d([`*`(s, `*`(cos(t))), `*`(s, `*`(sin(t))), s], s = 0 .. 4, t = 0 .. `+`(`*`(3, `*`(Pi))));  

Plot_2d
 

Parametrized Cylinder `+`(`*`(`^`(y, 2)), `*`(`^`(z, 2))) = 16;  

 

plot3d([s, `+`(`*`(4, `*`(cos(t)))), `+`(`*`(4, `*`(sin(t))))], s = -6 .. 5, t = 0 .. `+`(`*`(2, `*`(Pi))), scaling = constrained);  

Plot_2d
 

`+`(`*`(`^`(y, 2)), `*`(`^`(z, 2))) = 16;  

`+`(`*`(`^`(y, 2)), `*`(`^`(z, 2))) = 16 (12)