3149.sagews

x,y,z=var('x,y,z')
P = Graphics()

P += line( [(0,0,-1/3),(0,0,1)],thickness=4,color="red" )
P += text3d( "z", (0,0,5/4))
P += line( [(0,-1,0),(0,1,0)],thickness=4,color="green" )
P += text3d( "y", (0,5/4,0))
P += line( [(-1,0,0),(1,0,0)],thickness=4,color="yellow" )
P += text3d( "x", (5/4,0,0))

n=2

P += line( [(0,0,0),(-1/2,0,0)],thickness=n,color="black")
P += line( [(0,0,0),(0,1/3,0)],thickness=n,color="black")
P += line( [(0,0,0),(0,0,2/3)],thickness=n,color="black")

P += line( [(-1/2,0,0),(-1/2,1/3,0)],thickness=n,color="black")
P += line( [(-1/2,0,0),(-1/2,0,2/3)],thickness=n,color="black")

P += line( [(0,1/3,0),(-1/2,1/3,0)],thickness=n,color="black")
P += line( [(0,1/3,0),(0,1/3,2/3)],thickness=n,color="black")

P += line( [(0,0,2/3),(-1/2,0,2/3)],thickness=n,color="black")
P += line( [(0,0,2/3),(0,1/3,2/3)],thickness=n,color="black")

P += line( [(0,1/3,2/3),(-1/2,1/3,2/3)],thickness=n,color="black")
P += line( [(-1/2,0,2/3),(-1/2,1/3,2/3)],thickness=n,color="black")
P += line( [(-1/2,1/3,0),(-1/2,1/3,2/3)],thickness=n,color="black")

P += point3d((-1/2,1/3,2/3),size=10,color="blue")


#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,frame=false,aspect_ratio=[1,1,1])

Caption: The coordinates of a point $P$ in 3-space are determined by a rectangular solid whose faces are parallel to the $xy$, $xz$, and $yz$ planes and which has the origin and $P$ diagonally opposite.

x,y,z=var('x,y,z')
P = Graphics()

P += line( [(0,0,-1/3),(0,0,1)],thickness=2,color="black" )
P += text3d( "z", (0,0,5/4))
P += line( [(0,-1,0),(0,1,0)],thickness=2,color="black" )
P += text3d( "y", (0,5/4,0))
P += line( [(-1,0,0),(1,0,0)],thickness=2,color="black" )
P += text3d( "x", (5/4,0,0))

n=2

P += line( [(0,0,0),(-1/2,0,0)],thickness=n,color="black")
P += line( [(0,0,0),(0,1/3,0)],thickness=n,color="black")
P += line( [(0,0,0),(0,0,2/3)],thickness=n,color="black")

P += line( [(-1/2,0,0),(-1/2,1/3,0)],thickness=n,color="black")
P += line( [(-1/2,0,0),(-1/2,0,2/3)],thickness=n,color="black")

P += line( [(0,1/3,0),(-1/2,1/3,0)],thickness=n,color="black")
P += line( [(0,1/3,0),(0,1/3,2/3)],thickness=n,color="black")

P += line( [(0,0,2/3),(-1/2,0,2/3)],thickness=n,color="black")
P += line( [(0,0,2/3),(0,1/3,2/3)],thickness=n,color="black")

P += line( [(0,1/3,2/3),(-1/2,1/3,2/3)],thickness=n,color="black")
P += line( [(-1/2,0,2/3),(-1/2,1/3,2/3)],thickness=n,color="black")
P += line( [(-1/2,1/3,0),(-1/2,1/3,2/3)],thickness=n,color="black")

P += point3d((-1/2,1/3,2/3),size=10,color="blue")

P += line( [(0,0,0),(-1/2,1/3,2/3)],thickness=2*n,color="red")
P += line( [(0,1/3,0),(-1/2,1/3,2/3)],thickness=2*n,color="green")

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,frame=false,aspect_ratio=[1,1,1])

Caption: The distance from the origin to $P$ can be determined by applying the Pythagorean theorem twice; once to relate the lengths of the red and green lines and once to find the length of the green line.

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (t*cos(s),t*sin(s),0), (s,0,pi/2),(t,1,2),opacity=0.8)
P+=parametric_plot3d( (t*cos(s),0,t*sin(s)), (s,0,pi/2),(t,1,2),opacity=0.8)
P+=parametric_plot3d( (0,t*cos(s),t*sin(s)), (s,0,pi/2),(t,1,2),opacity=0.8)

P+=parametric_plot3d( (sin(s)*cos(t),sin(s)*sin(t),cos(s)), (s,0,pi/2),(t,0,pi/2),opacity=0.8)

P+=parametric_plot3d( (2*sin(s)*cos(t),2*sin(s)*sin(t),2*cos(s)), (s,0,pi/2),(t,0,pi/2),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: The region which satisfies the inequalities $1\leq x^2+y^2+z^2 \leq 4$ and $x,y,z\geq 0$

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (cos(s),sin(s),t), (s,0,2*pi),(t,-1,1),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In cylindrical coordinates, the solutions to $r=1$.

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (s*cos(pi/4),s*sin(pi/4),t), (s,-sqrt(2),sqrt(2)),(t,-1,1),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In cylindrical coordinates, the solutions to $\theta=\pi/4$

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (s,t,1), (s,-1,1),(t,-1,1),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In cylindrical coordinates, the solutions to $z=1$

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (sin(s)*cos(t),sin(s)*sin(t),cos(s)), (s,0,2*pi),(t,0,2*pi),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In spherical coordinates, the solutions to $\rho=1$

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (s*cos(pi/4),s*sin(pi/4),t), (s,-sqrt(2),sqrt(2)),(t,-1,1),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In spherical coordinates, the solutions to $\theta=\pi/4$

x,y,z,s,t=var('x,y,z,s,t')
P = Graphics()

P+=parametric_plot3d( (s*sin(pi/4)*cos(t),s*sin(pi/4)*sin(t),s*cos(pi/4)), (s,-1,1),(t,0,2*pi),opacity=0.8)

#P.show(figsize=8,frame=false,aspect_ratio=[1,1,1],viewer="tachyon")
P.show(figsize=8,aspect_ratio=[1,1,1])

Caption: In spherical coordinates, the solutions to $\phi=\pi/4$