4126.sagews

• Vector functions of a curve have 1 parameter, i.e. 1 variable (regardless of the number of components).
• Vector functions can have 1, 2, 3 or more components = dimension. Components are separated by commas.
• In SAGE, use DOUBLE parentheses to define a vector function!
• Top section: define and plot vector (parametric) functions of curves.
• Below are commands: extract components r[0], evaluate r(t=3), magnitude abs(r) norm(r), derivatives diff(r,t)

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var ('t')
r=vector((t,t^2))
parametric_plot(r,(t,-2,2), figsize=2)

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Notice that parametric_plot is 2d or 3d depending on what it is plotting. (No separate command.)

var ('t')
r=vector((sin(t),cos(t),t)) 
parametric_plot(r,(t,-2*pi,2*pi),color='purple',thickness=3, aspect_ratio=(3,3,1))

view(r)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\sin\left(t\right),\,\cos\left(t\right),\,t\right)

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Extract component functions.

To project onto the x-y plane, define a new vector function using the first 2 components of 3d vector function (numbering starts at 0).

var ('t')
r=vector((sin(t),cos(t),t))
r2=vector((r[0],r[1]))

p2=parametric_plot(r2,(t,-2*pi,2*pi),rgbcolor=(1,0,0),thickness=2, aspect_ratio=1)
show(p2, figsize=3)

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You cannot just force a 3d plot of 2d curve with aspect_ratio. (You will get an error as 1- below.)

In order to plot the 2d curve, you need to also plot an actual 3d curve. If you don't want to see it, makes its opacity=0. (See 2-below.)

var ('t')
r=vector((sin(t),cos(t),t))
r2=vector((r[0],r[1]))
p2_3=parametric_plot(r2,(t,-2*pi,2*pi),color='red',thickness=5, aspect_ratio=(3,3,1))
show(p2_3)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_15.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("dmFyICgndCcpCnI9dmVjdG9yKChzaW4odCksY29zKHQpLHQpKQpyMj12ZWN0b3IoKHJbMF0sclsxXSkpCnAyXzM9cGFyYW1ldHJpY19wbG90KHIyLCh0LC0yKnBpLDIqcGkpLGNvbG9yPSdyZWQnLHRoaWNrbmVzcz01LCBhc3BlY3RfcmF0aW89KDMsMywxKSkKc2hvdyhwMl8zKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpJik_OI/___code___.py", line 7, in <module> exec compile(u'show(p2_3)' + '\n', '', 'single') File "", line 1, in <module> File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/misc/functional.py", line 1487, in show return x.show(*args, **kwds) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/misc/decorators.py", line 381, in wrapper return func(*args, **kwds) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/plot/plot.py", line 1726, in show self.save(**kwds) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/misc/decorators.py", line 381, in wrapper return func(*args, **kwds) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/plot/plot.py", line 2453, in save figure = self.matplotlib(**options) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/plot/plot.py", line 1945, in matplotlib ymin=ymin, ymax=ymax) File "/home/sage/sage-4.7/local/lib/python2.6/site-packages/sage/plot/plot.py", line 3886, in adjust_figsize_for_aspect_ratio r = max(aspect_ratio * (xmax - xmin)/(ymax-ymin), 0.001) TypeError: can't multiply sequence by non-int of type 'float'

var ('t')
r=vector((sin(t),cos(t),t))
r2=vector((r[0],r[1]))
p3=parametric_plot(r,(t,-2*pi,2*pi),opacity=0)
p2_3=parametric_plot(r2,(t,-2*pi,2*pi),color='red',thickness=5, aspect_ratio=(3,3,1))
show(p3+p2_3)

Click and drag to rotate this to see that we have projected the helix onto the circle.

var ('t')
r=vector((sin(t),cos(t),t))
r2=vector((r[0],r[1]))
p3 = parametric_plot(r,(t,-2*pi,2*pi),opacity=0.5,thickness=2,aspect_ratio=(3,3,1))
p2_3 = parametric_plot(r2,(t,-2*pi,2*pi),color='red',thickness=3)
var ('x y z')
px0y= implicit_plot3d(z==0,(x,-1,1),(y,-1,1),(z,-2*pi,2*pi), color='grey', opacity=0.3)
show(p3+p2_3+px0y)

var ('t')
r=vector((sin(t),cos(t),t))
r2=vector((r[0],r[1],3))
p2_b = parametric_plot(r2,(t,-2*pi,2*pi),color='red',thickness=5)
show(p2_b)

var ('t')
r=vector((sin(t),cos(t),t))
m=sqrt(r[0]^2+r[1]^2+r[2]^2)
view(m)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{t^{2} + \sin\left(t\right)^{2} + \cos\left(t\right)^{2}}

m2=abs(r)
view(m2)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{t^{2} + \sin\left(t\right)^{2} + \cos\left(t\right)^{2}}

m3=r.norm()
view(m3)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left| t \right|}^{2} + {\left| \sin\left(t\right) \right|}^{2} + {\left| \cos\left(t\right) \right|}^{2}}

m3a=norm(r)
view(m3a)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left| t \right|}^{2} + {\left| \sin\left(t\right) \right|}^{2} + {\left| \cos\left(t\right) \right|}^{2}}

rprime=diff(r,t)
view(rprime)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\cos\left(t\right),\,-\sin\left(t\right),\,1\right)

rprime2=r.diff(t)
view(rprime2)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\cos\left(t\right),\,-\sin\left(t\right),\,1\right)

ds=norm(rprime)
view(ds)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{{\left| -\sin\left(t\right) \right|}^{2} + {\left| \cos\left(t\right) \right|}^{2} + 1}

ds(t=3.0)

\newcommand{\Bold}[1]{\mathbf{#1}}1.41421356237310

n(ds(t=3))

\newcommand{\Bold}[1]{\mathbf{#1}}1.41421356237310

var ('t')
r=vector((sin(t),cos(t)))
m=sqrt(r[0]^2+r[1]^2)
view(m)

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\sin\left(t\right)^{2} + \cos\left(t\right)^{2}}

m(t=3)

sqrt(sin(3)^2 + cos(3)^2)

n(m(t=3))

1.00000000000000