# Answers to Maple Project 1/ mth243 

# Problem 1(a)

with(plots):
x:=log(t+1);y:=cos(t);z:=sin(t);
spacecurve([x,y,z],t=0..25,axes=normal,color=red,orientation=[60,40],labels=["x","y","z"]); 
# The motion is spiral along the X-axis, whose rotation (of constant radius)
# gets tighter or for every rotation the ditance from the last motion decreases 

# Problem 1(b)
int(diff(x,t)^2+diff(y,t)^2+diff(z,t)^2, t=0..5/12);
evalf(%); # the command evalf is needed only if you use "Int" for integration
# The object has traveled just about 54 centimeters over a time period of 25
# seconds = 5/12 min

# Problem 2(a)

plot3d([4*cos(theta)+2*cos(psi)*cos(theta),4*sin(theta)+2*cos(psi)*sin(psi),2*sin(psi)],psi=0..2*Pi,theta=0..Pi,axes=framed,labels=["x","y","z"],scaling=constrained);

# Problem 2(b)

plot3d([4*cos(theta)+2*cos(psi)*cos(theta),4*sin(theta)+2*cos(psi)*sin(psi),2*sin(psi)],psi=0..Pi,theta=0..2*Pi,axes=framed,labels=["x","y","z"],scaling=constrained);
# The graphs in parts (a) and (b) are different: part(b) is half of donuts. 


# Problem 3(a)

plot3d([(s^2+1)*cos(theta),(s^2+1)*sin(theta),s],theta=0..2*Pi,s=-2..2,axes=framed,labels=["x","y","z"],orientation=[52,76]);

# Problem 3(b)

plot3d([5*sin(psi)*cos(theta),5*sin(psi)*sin(theta),5*cos(psi)],psi=-Pi..0,theta=-Pi..Pi/2,axes=framed,labels=["x","y","z"],scaling=constrained);

# Problem 3(c)
plot3d([s,10*(s/10)^2-10,t],s=-10..10,t=-10..0,axes=framed,labels=["x","y","z"],scaling=constrained);

## ===== Other solutions: 
# Problem 3(a)
plot3d([(2.5*cos(t)-3.5)*cos(s),(2.5*cos(t)-3.5)*sin(s),t],s=0..2*Pi,t=-2..2,axes=framed,labels=["x","y","z"],orientation=[52,76]);
plot3d([sqrt(s^2+1)*cos(theta),sqrt(s^2+1)*sin(theta),s],theta=0..2*Pi,s=-2..2,axes=framed,labels=["x","y","z"],orientation=[52,76]);
plot3d([2*sec(s)*cos(theta),2*sec(s)*sin(theta),tan(s)],theta=-Pi..Pi,s=-Pi/3..Pi/3,axes=framed,labels=["x","y","z"],orientation=[52,76]);

# Problem 3(b)
plot3d([sqrt(4*4-s*s)*cos(t),sqrt(4*4-s*s)*sin(t),s],s=-4..4,t=0..3*Pi/2,axes=framed,labels=["x","y","z"],scaling=constrained);

# Problem 3(c)
plot3d([-10*cos(s),-10*sin(s),t],s=0..Pi,t=-10..0,axes=framed,labels=["x","y","z"],orientation=[52,76]);
plot3d([10*sin(s),10*cos(s),t],t=0..10,s=0..Pi,axes=normal,color=red,orientation=[52,76],labels=["x","y","z"]);
plot3d([x,(1/10)*x*x-10,z],x=-10..10,z=-10..0,axes=normal,color=red,orientation=[52,76],labels=["x","y","z"]);
plot3d([10*cos(x),-10*sin(x)*sin(x),-10*sin(t)],x=0..2*Pi,t=0..Pi,axes=normal,color=red,orientation=[52,76],labels=["x","y","z"]);
plot3d([x^2,-5*x-5,t],t=-10..0,x=-1..1,axes=framed,labels=["x","y","z"]);