A plane. Or, with more details, NotebookDirectory. C:\Dropbox\Work\myweb\Courses\Math_pages\Math_225\

Transcription

1 In[1]:= NotebookDirectory Out[1]= C:\Dropbox\Work\myweb\Courses\Math_pages\Math_5\ A plane Given a point in R 3 (below it is vr) and two non-collinear vectors (below uu and vv) the parametric equation of the plane which goes through the given point and is parallel to the given vectors is illustrated below. In[]:= vr 1,, 1; uu, 3, ; vv 1,, 3; ParametricPlot3Dvr s uu t vv, s, 13, 13, t, 13, 13, PlotRange 5, 5, 5, 5, 5, 5 Out[3]= In[4]:= Or, with more details,? Mesh Mesh is an option for Plot3D, DensityPlot and other plotting functions that specifies what mesh should be drawn.

2 Surface_15.nb In[5]:= vr 1,, 1; uu, 3, ; vv 1,, 3; Show ParametricPlot3Dvr s uu t vv, s, 5, 5, t, 5, 5, PlotStyle Opacity.75, Mesh Range1, 1, 1, Range1, 1, 1, MeshStyle Thickness.3, Purple, Thickness.3, Blue, PlotRange 5, 5, 5, 5, 5, 5, AxesLabel x, y, z, Graphics3DBlack, Thickness.15, Arrow,,, vr, PointSize.3, Pointvr, Blue, Thickness.1, Arrowvr, vr uu, Purple, Thickness.1, Arrowvr, vr vv Out[6]= A random surface In general, a surface in R 3 is given by a triple of equations: x f s, t, y gs, t, z hs, t. Below we came up with three random formulas and plotted the resulting surface.

3 Surface_15.nb 3 In[7]:= ParametricPlot3Ds t, s Expt, Cost Logs, s,, 5, t, 13, 13, PlotRange 3, 3,,, 1, 1, PlotPoints 5, 5, AxesLabel x, y, z Out[7]= A sphere However, it is more interesting to find parametric equations for familiar surfaces. The most common parametric equations for a unit sphere centered at the origin are equations obtained from the spherical coordinates.

4 4 Surface_15.nb In[8]:= ParametricPlot3DCos Sin, Sin Sin, Cos,,, Pi,,, Pi, PlotRange 1., 1., 1., 1., 1., 1. Out[8]= In[9]:= sph1 ParametricPlot3DCos Sin, Sin Sin, Cos,,, Pi,,, Pi, PlotStyle Opacity.5, PlotRange 1., 1., 1., 1., 1., 1. Out[9]= Below, I will illustrate how you can view the unit sphere as a union of horizontal circles with varying radii (blue) and

5 Surface_15.nb 5 also as a union of vertical semi-circles (purple). In[1]:= Manipulate ShowParametricPlot3D,, Coss Sins Cos, Sin,,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 1., 1., 1., 1., 1., 1., ParametricPlot3DCos,, 1 Sin Cost, Sint,,,, Pi, PlotStyle Thickness.1, Purple, PlotRange 1., 1., 1., 1., 1., 1., sph1, s, Pi,, Pi, t,, Pi s t Out[1]= The horizontal circles with varying radii (blue) make up the sphere:

6 6 Surface_15.nb In[11]:= Manipulate ShowTableParametricPlot3D,, Cossr Sinsr Cos, Sin,,,, Pi, PlotStyle Thickness.7, Blue, PlotRange 1., 1., 1., 1., 1., 1., sr,, s,.1, s,, Pi s Out[11]= The vertical semi-circles (purple) make up the sphere:

7 Surface_15.nb 7 In[1]:= Manipulate ShowTableParametricPlot3D,, Cos Sin Costr, Sintr,,,, Pi, PlotStyle Thickness.7, Purple, PlotRange 1., 1., 1., 1., 1., 1., tr,, t, Pi 3, t,, Pi t Out[1]= Next, I will show how to place a decoration on the sphere. Assume that at t we start from the North pole of the sphere, that is the point,, 1. Than in we proceed downwards and after circling the sphere three times we end up at the South pole, that is the point,, 1. Our height could be represented by the function Cost 6. The other coordinates are as follows below.

8 8 Surface_15.nb In[13]:= dec ParametricPlot3DSin t 6 Cost, Sin t 6 Sint, Cos t 6, t,, 6 Pi, PlotStyle Thickness.1, Red Out[13]=

9 Surface_15.nb 9 In[14]:= Showsph1, dec Out[14]= A vase In the previous section we have seen that the sphere can be viewed as a union of horizontal circles varying radii. Similarly, a vase can be viewed as a union of horizontal circles varying radii. To illustrate this, we have to come up with a formula for the radius at the level z.

10 1 Surface_15.nb In[15]:= Manipulate ParametricPlot3D Sinz Cos, Sinz Sin, z,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 3, 3, 3, 3,.1, Pi, z,,, Pi z 6 Out[15]= 4

11 Surface_15.nb 11 In[16]:= Manipulate Show ParametricPlot3D Sinz Cos, Sinz Sin, z,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 3, 3, 3, 3,.1, Pi, ParametricPlot3D Sins Cos, Sins Sin, s,,, Pi, s,, z, PlotRange 3, 3, 3, 3,.1, Pi, z,.1,.1, Pi z Out[16]= We can also view the vase as a union of the graphs of the function Sinz in the vertical planes that contain the z- axis.

12 1 Surface_15.nb In[17]:= Manipulate Show ParametricPlot3D Sinz Cos, Sin, z,, 1, z,, Pi, PlotStyle Thickness.1, Purple, PlotRange 3, 3, 3, 3,.1, Pi, Graphics3DThickness.1, Arrow,,, Cos, Sin,, Thickness.1, Arrow,,,,, 1, Line,,, 1 Cos, Sin,, Line,,, 1,, 1,,,, Pi 6 Out[17]= 4 In[18]:= Manipulate ShowParametricPlot3D Sinz Cos, Sin, z,, 1, z,, Pi, PlotStyle Thickness.1, Purple, PlotRange 3, 3, 3, 3,.1, Pi, ParametricPlot3D Sinz Cost, Sint, z,, 1, z,, Pi, t,,, PlotRange 3, 3, 3, 3,.1, Pi,,.1,.1, Pi

13 Surface_15.nb 13 Out[18]=

14 14 Surface_15.nb Out[71]= Let us now put a decoration on this vase.

15 Surface_15.nb 15 In[19]:= vase ParametricPlot3D Sinz Cost, Sint, z,, 1, z,, Pi, t,, Pi, PlotStyle Opacity.6, PlotRange 3, 3, 3, 3,.1, Pi Out[19]=

16 16 Surface_15.nb In[]:= decv ParametricPlot3D Sin t 3 Cost, Sint, t,, 1, 3 t,, 6 Pi, PlotStyle Thickness.1, Red 6 Out[]= 4

17 Surface_15.nb 17 In[1]:= Showvase, decv Out[1]= A torus The same principle that we used for the sphere and the vase with different radii lead to a torus. At the level z Sins let the radius of the circle be Coss. We start from s and the radius 1. As s increases the level Sins increases and the radius increases. At s we are at the level 1 and the radius is. As s increases further, the level goes down, but the radius continuous to increase. At s the level is again, but the radius is 3. Think what is happening while watching the manipulation below.

18 18 Surface_15.nb In[]:= Manipulate ShowTableParametricPlot3D,, Sinsr Cossr Cos, Sin,,,, Pi, PlotRange 3., 3., 3., 3., 1., 1., sr,, s,.1, s,, Pi s Out[]=

19 Surface_15.nb 19 In[3]:= Manipulate ParametricPlot3D Cos 1,, Sin, 1, Cost Cos 1,, Sin, 1, Sint,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 3, 3, 3, 3, 1.1, 1.1,,,, Pi Out[3]=

20 Surface_15.nb In[4]:= Manipulate Show ParametricPlot3D Cos 1,, Sin, 1, Cost Cos 1,, Sin, 1, Sint,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 3, 3, 3, 3, 1.1, 1.1, ParametricPlot3D Coss 1,, Sins, 1, Cost Coss 1,, Sins, 1, Sint,, 1, t,, Pi, s,,, PlotRange 3, 3, 3, 3, 1.1, 1.1,,.1,.1, Pi Out[4]=

21 Surface_15.nb 1 In[5]:= Manipulate ParametricPlot3D Coss Cos 1,, Sin, 1, Sins,, 1,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 3, 3, 3, 3, 1.1, 1.1, s,,, Pi s Out[5]=

22 Surface_15.nb In[6]:= Manipulate Show ParametricPlot3D Coss Cos 1,, Sin, 1, Sins,, 1,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 3, 3, 3, 3, 1.1, 1.1, ParametricPlot3D Cossr Cos 1,, Sin, 1, Sinsr,, 1,,, Pi, sr,, s, PlotStyle Thickness.1, PlotRange 3, 3, 3, 3, 1.1, 1.1, s,.1,.1, Pi s Out[6]= Or, changing the direction of movement of the circles

23 Surface_15.nb 3 In[7]:= Manipulate Show ParametricPlot3D Coss Cos 1,, Sin, 1, Sins,, 1,,, Pi, PlotStyle Thickness.1, Blue, PlotRange 3, 3, 3, 3, 1.1, 1.1, ParametricPlot3D Cossr Cos 1,, Sin, 1, Sinsr,, 1,,, Pi, sr,, s, PlotStyle Thickness.1, PlotRange 3, 3, 3, 3, 1.1, 1.1, s, 4 Pi,.1, Pi 3 s Out[7]= Recall the cardioid that we mentioned when we talked about parametrized curves.

24 4 Surface_15.nb In[8]:= Cleart, rc; rct_ : 1 Cost Cost, Sint; Graphics, Thick, Blue, LineTablercv, v,, Pi, Pi 18 Frame True, PlotRange.5,.5, 1.5, 1.5, AspectRatio Automatic, GridLines, GrayLevel.5, Dashing.1,.1 & Range1, 1,, GrayLevel.5, Dashing.1,.1 & Range1, Out[3]= This is more detailed way of writing its equation

25 Surface_15.nb 5 In[31]:= ParametricPlot 1 Cost Cost 1, 1 Cost Sint, 1, t,, Pi, Frame True, PlotRange.5,.5, 1.5, 1.5, AspectRatio Automatic, GridLines, GrayLevel.5, Dashing.1,.1 & Range1, 1,, GrayLevel.5, Dashing.1,.1 & Range1, Out[31]= In[3]:= I want to use this cardioid to make a torus from it. I will first move it away from the origin and flip it around ParametricPlot 3 1, 1 Cost Cost 1, 1 Cost Sint, 1, t,, Pi, Frame True, PlotRange.5, 4.5, 1.5, 1.5, AspectRatio Automatic, GridLines, GrayLevel.5, Dashing.1,.1 & Range1, 1,, GrayLevel.5, Dashing.1,.1 & Range1, Out[3]= Now, I will place this cardioid vertically and rotate it around the z-axis.

26 6 Surface_15.nb In[33]:= Manipulate ParametricPlot3D3 1 Cost Cost Cos, Sin, 1 Cost Sint,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 4, 4, 4, 4, 1.5, 1.5,,,, Pi Out[33]= Finally, In[34]:= ParametricPlot3D3 1 Cost Cost Cos, Sin, 1 Cost Sint,, 1, t,, Pi,,, Pi, PlotRange 4, 4, 4, 4, 1.5, 1.5 Out[34]= Since the above picture is not too interesting I will try rotating the cardioid as changes

27 Surface_15.nb 7 In[35]:= Manipulate ParametricPlot3D 3 Cos, Sin, 1 Cost Cost Cos Cos, Sin, Sin,, 1 1 Cost Sint Sin Cos, Sin, Cos,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 5.5, 5.5, 5.5, 5.5,,,,,, Pi Out[35]=

28 8 Surface_15.nb In[36]:= Manipulate Show ParametricPlot3D3 Cos, Sin, 1 Cost Cost Cos Cos, Sin, Sin,, 1 1 Cost Sint Sin Cos, Sin, Cos,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 5.5, 4.5, 5.5, 4.5,,, ParametricPlot3D3 Coss, Sins, 1 Cost Cost Coss Coss, Sins, Sins,, 1 1 Cost Sint Sins Coss, Sins, Coss,, 1, t,, Pi, s,.1,, Mesh False, PlotRange 5.5, 4.5, 5.5, 4.5,,,,,, Pi Out[36]=

29 Surface_15.nb 9 In[37]:= ParametricPlot3D 3 Cos, Sin, 1 Cost Cost Cos Cos, Sin, Sin,, 1 1 Cost Sint Sin Cos, Sin, Cos,, 1, t,, Pi,,, Pi, Mesh False, PlotRange 5.5, 4.5, 5.5, 4.5,, Out[37]= Let us rotate the cardioid several times

30 3 Surface_15.nb In[38]:= nn ; Manipulate ParametricPlot3D3 Cos, Sin, 1 Cost Cost Cosnn Cos, Sin, Sinnn,, 1 1 Cost Sint Sinnn Cos, Sin, Cosnn,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 5.5, 5.5, 5.5, 5.5,,,,,, Pi Out[38]=

31 Surface_15.nb 31 In[39]:= nn 3; ParametricPlot3D3 Cos, Sin, 1 Cost Cost Cosnn Cos, Sin, Sinnn,, 1 1 Cost Sint Sinnn Cos, Sin, Cosnn,, 1, t,, Pi,,, Pi, PlotStyle Opacity.6, PlotPoints 5, 5, Mesh False, PlotRange 5.5, 5.5, 5.5, 5.5,, Out[39]=

32 3 Surface_15.nb In[4]:= nn 3; Manipulate Show ParametricPlot3D3 Cos, Sin, 1 Cost Cost Cosnn Cos, Sin, Sinnn,, 1 1 Cost Sint Sinnn Cos, Sin, Cosnn,, 1, t,, Pi, PlotStyle Thickness.1, Purple, PlotRange 5.5, 5.5, 5.5, 5.5,,, ParametricPlot3D3 Coss, Sins, 1 Cost Cost Cosnn s Coss, Sins, Sinnn s,, 1 1 Cost Sint Sinnn s Coss, Sins, Cosnn s,, 1, t,, Pi, s,.1,, PlotStyle Opacity.9, PlotPoints 5, 5, Mesh False, PlotRange 5.5, 5.5, 5.5, 5.5,,,,,, Pi Out[4]=

33 Surface_15.nb 33 Making a surfaces starting from a curve In[41]:= Starting from a circle Show ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, PlotRange,,,,, Out[41]= 1 1 At each point of this circle we can place a vertical line. 1

34 34 Surface_15.nb In[4]:= t 1; Show ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3DCost, Sint,,, s, s, 3, 3, PlotStyle Cyan, Thickness.1, PlotRange,,,,, Out[4]= 1 1 1

35 Surface_15.nb 35 In[43]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3DCost, Sint,,, s, s, 3, 3, PlotStyle Cyan, Thickness.1, PlotRange,,,,,, t,, Pi t 1 1 Out[43]=

36 36 Surface_15.nb In[44]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3DCost, Sint,,, s, s, 3, 3, t,, t, PlotRange,,,,,, t,.1, Pi t Out[44]= However, instead of using a vertical line we could go in any direction, Say

37 Surface_15.nb 37 In[45]:= vv 1,, 3; ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3DCost, Sint, s vv, s, 3, 3, t,, t, PlotRange,,,,,, t,.1, Pi t Out[45]= A different way of getting interesting surfaces is to move the original circle.

38 38 Surface_15.nb In[46]:= Show ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, 1 Cost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, PlotRange,,,,, Out[46]= 1 1 1

39 Surface_15.nb 39 In[47]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, s Cost, Sint,, t,, Pi, s,, h, PlotRange,,,,, 4, h,.1, 4 h Out[47]= However, it is not hard to get much more imaginative by moving the center as we go up

40 4 Surface_15.nb In[48]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3DCoss, Sins, s Cost, Sint,, t,, Pi, s,, h Pi, PlotRange,,,,, 4, h,.1, 4 Pi h Out[48]= Another twist that we can introduce is to vary the radius as we go up

41 Surface_15.nb 41 In[49]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D Coss, Sins,, s 1 Sins Cost, Sint,, t,, Pi, s,, h Pi PlotRange 3, 3, 3, 3,, 4, h,.1, 4 Pi h Out[49]= This idea of varying the radius leads to a familiar surface

42 4 Surface_15.nb In[5]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, s s Cost, Sint,, t,, Pi, s,, h, PlotRange 3, 3, 3, 3,, 4, h,.1, 4 h Out[5]=

43 Surface_15.nb 43 In[51]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, s Sins Cost, Sint,, t,, Pi, s,, h, PlotRange 3, 3, 3, 3,, 4, h,.1, Pi h Out[51]=

44 44 Surface_15.nb In[5]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, Coss Sins Cost, Sint,, t,, Pi, s,, h, PlotRange 1.1, 1.1, 1.1, 1.1, 1.1, 1.1, h,.1, Pi h Out[5]=

45 Surface_15.nb 45 In[53]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, Sins Coss Cost, Sint,, t,, Pi, s,, h, PlotRange 3.1, 3.1, 3.1, 3.1, 1.1, 1.1, h,.1, Pi h Out[53]=

46 46 Surface_15.nb In[54]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D,, Coss,, Sins, s,, Pi, PlotStyle Cyan, Thickness.1, PlotRange 3.1, 3.1, 3.1, 3.1, 1.1, 1.1, h,.1, Pi h Out[54]=

47 Surface_15.nb 47 In[55]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D 1,, Coss 1,, Sins,, 1, s,, Pi, PlotStyle Cyan, Thickness.1, PlotRange 3.1, 3.1, 3.1, 3.1, 1.1, 1.1, h,.1, Pi h Out[55]=

48 48 Surface_15.nb In[56]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D Costt, Sintt, Coss Costt, Sintt, Sins,, 1, s,, Pi, PlotStyle Cyan, Thickness.1, PlotRange 3.1, 3.1, 3.1, 3.1, 1.1, 1.1, tt,, Pi tt Out[56]=

49 Surface_15.nb 49 In[57]:= ManipulateShow ParametricPlot3DCost, Sint,, t,, Pi, PlotStyle Blue, Thickness.1, ParametricPlot3D Costt, Sintt, Coss Costt, Sintt, Sins,, 1, s,, Pi, PlotStyle Cyan, Thickness.1, ParametricPlot3D Cost, Sint, Coss Cost, Sint, Sins,, 1, s,, Pi, t,, tt, PlotRange 3.1, 3.1, 3.1, 3.1, 1.1, 1.1, tt,.1, Pi tt Out[57]=