The Skier's Path

Adapted from the Maple 8 Getting Started Guide

A skier is close to the top of a mountain.  She wants to take the steepest path down.  This worksheet investigates finding this path.

The height at a point on the mountain as a function of (x,y) is given by

, in thousands of feet.

Picture of the Mountain

>

x:='x';

>

y:='y';

>	with(plots):plot3d(3/((1+x^2+y^2)*(1/4+((x+1)^2)/2+((y+2)^2)/2)),x=-4..3,y=-4..3,numpoints=2500,axes=boxed,orientation=[35,60],contours=20);

>	f:=3/((1+x^2+y^2)*(1/4+((x+1)^2)/2+((y+2)^2)/2));

>	fx:=diff(f,x);

>	fy:=diff(f,y);

Approximating the Peak of the Mountain

>	HighPt:=fsolve({fx=0,fy=0},{x,y},{x=-2..0,y=-2..0});

>	assign(%);

Identifying a Point Near the Peak of the Mountain (Starting Point)

>	x1:=x+0.05;

>	y1:=y+0.05;

>

x:='x';

>

y:='y';

>	z1:=eval(f,{x=x1,y=y1});

Contour Plot

>	contourplot(f,x=-3..2,y=-4..2,contours=8,filled=true);

The Mountain with a Starting Point

>	start1:=pointplot3d([x1,y1,z1],symbol=cross,symbolsize=50,color=yellow):

>	Mountain:=plot3d(3/((1+x^2+y^2)*(1/4+((x+1)^2)/2+((y+2)^2)/2)),x=-3..2,y=-4..2,numpoints=2500,axes=boxed,orientation=[35,60],contours=20):

>	display(Mountain,start1);

Approximating a Vector at the Starting Point in the Direction of Steepest Descent

(Note:  The x and y Components Are in the Direction of the Negative Gradient and the z Component Is Computed to Put the Endpoint of the Vector on the Surface of the Mountain.)

>	xx1:=eval(-fx,{x=x1,y=y1});

>	yy1:=eval(-fy,{x=x1,y=y1});

>	zz1:=eval(f,{x=x1+xx1,y=y1+yy1})-z1;

>	DirVec1:=arrow(<x1,y1,z1>,<xx1,yy1,zz1>,color=red,width=0.06):

>	display(Mountain,start1,DirVec1);

Approximating a Vector in the Direction of Steepest Descent at the Endpoint of the Previous "Steepest Descent" Vector

>	xx2:=eval(-fx,{x=x1+xx1,y=y1+yy1})/4;

>	yy2:=eval(-fy,{x=x1+xx1,y=y1+yy1})/4;

>	zz2:=eval(f,{x=x1+xx1+xx2,y=y1+yy1+yy2})-z1-zz1;

>	DirVec2:=arrow(<x1+xx1,y1+yy1,z1+zz1>,<xx2,yy2,zz2>,color=red,width=0.06):

>	display(Mountain,start1,DirVec1,DirVec2);

Approximating the Path the Skier Would Take Down the Mountain

>	g:=eval(f,{x=P[1],y=P[2]});

>	gx:=eval(fx,{x=P[1],y=P[2]});

>	gy:=eval(fy,{x=P[1],y=P[2]});

>	point3d:=Array(1..25);

>	route3d:=Array(1..25);

>	timestep:=0.1;

>	point3d[1]:=<x1,y1,z1>;

>	for i from 1 to 24 do route3d[i]:=LinearAlgebra[Normalize](eval(<-gx,-gy,0>,P=point3d[i])); point3d[i+1]:=eval(<P[1],P[2],g>,P=point3d[i]+timestep*route3d[i]); end do:

>	listpoints3d:=[seq(convert(point3d[i],list),i=1..25)]:

>	path3d1:=pointplot3d(listpoints3d,style=line,color=red,thickness=3):

>	display(Mountain,start1,path3d1);

The Path of the Skier Projected onto the Contour Plot

>	LevelContour:=contourplot(f,x=-3..2,y=-4..2,contours=8,filled=true):

>	gx:=eval(fx,{x=P[1],y=P[2]});

>	gy:=eval(fy,{x=P[1],y=P[2]});

>	point2d1:=Array(1..25);

>	route2d1:=Array(1..25);

>	timestep:=0.1;

>	point2d1[1]:=<x1,y1>;

>	for i from 1 to 24 do route2d1[i]:=LinearAlgebra[Normalize](eval(<-gx,-gy>,P=point2d1[i])); point2d1[i+1]:=eval(<P[1],P[2]>,P=point2d1[i]+timestep*route2d1[i]); end do:

>	listpoints2d1:=[seq(convert(point2d1[i],list),i=1..25)]:

>	path2d1:=pointplot(listpoints2d1,style=line,color=blue,thickness=3):

>	display(LevelContour,path2d1);

The Path of the Skier Projected onto the Contour Plot   Starting Just Slightly on the Other Side of the Peak

>

x:='x';

>

y:='y';

>	HighPt:=fsolve({fx=0,fy=0},{x,y},{x=-2..0,y=-2..0});

>	assign(%);

>	x1:=x-0.01;

>	y1:=y-0.01;

>

x:='x';

>

y:='y';

>	gx:=eval(fx,{x=P[1],y=P[2]});

>	gy:=eval(fy,{x=P[1],y=P[2]});

>	point2d2:=Array(1..25);

>	route2d2:=Array(1..25);

>	timestep:=0.1;

>	point2d2[1]:=<x1,y1>;

>	for i from 1 to 24 do route2d2[i]:=LinearAlgebra[Normalize](eval(<-gx,-gy>,P=point2d2[i])); point2d2[i+1]:=eval(<P[1],P[2]>,P=point2d2[i]+timestep*route2d2[i]); end do:

>	listpoints2d2:=[seq(convert(point2d2[i],list),i=1..25)]:

>	path2d2:=pointplot(listpoints2d2,style=line,color=blue,thickness=3):

>	display(LevelContour,path2d2);

Approximating the Peak of the Mountain   Again

>	HighPt:=fsolve({fx=0,fy=0},{x,y},{x=-2..0,y=-2..0});

>	assign(%);

Identifying the Point Again Near the Peak of the Mountain Used in the Contour Plot Above (Starting Point)

>	x1:=x-0.01;

>	y1:=y-0.01;

>

x:='x';

>

y:='y';

>	z1:=eval(f,{x=x1,y=y1});

>	start2:=pointplot3d([x1,y1,z1],symbol=cross,symbolsize=50,color=green):

>	g:=eval(f,{x=P[1],y=P[2]});

>	gx:=eval(fx,{x=P[1],y=P[2]});

>	gy:=eval(fy,{x=P[1],y=P[2]});

>	point3d:=Array(1..25);

>	route3d:=Array(1..25);

>	timestep:=0.1;

>	point3d[1]:=<x1,y1,z1>;

>	for i from 1 to 24 do route3d[i]:=LinearAlgebra[Normalize](eval(<-gx,-gy,0>,P=point3d[i])); point3d[i+1]:=eval(<P[1],P[2],g>,P=point3d[i]+timestep*route3d[i]); end do:

>	listpoints3d:=[seq(convert(point3d[i],list),i=1..25)]:

>	path3d2:=pointplot3d(listpoints3d,style=line,color=red,thickness=3):

Change the View of the Mountain

>	Mountain:=plot3d(3/((1+x^2+y^2)*(1/4+((x+1)^2)/2+((y+2)^2)/2)),x=-3.3..2,y=-4..2,numpoints=2500,axes=boxed,orientation=[-100,60],contours=20):

>	display(Mountain,start2,path3d2);

>	display(Mountain,start1,start2,path3d1,path3d2);

>	display(LevelContour,path2d1,path2d2);

>