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How to plot a circle in sage
A circle is defined either implicitly or parametrically. It is not defined explicitly.

Implicitly (see also implicit functions,implicit_plot)
Circle with center (0,0) and radius 3: x^2+y^2=9
To plot:
var ('x y')
implicit_plot(x^2+y^2==9,(x,-5,5),(y,-5,5))
Notice the "double" equals!

This works with off-center circles too except you will need to adjust your mins and maxes to get it.
Circle with center (2,-3) and radius \sqrt{5}: (x-2)^2+(y+3)^2=5
To plot:
var ('x y')
implicit_plot((x-2)^2+(y+3)^2==5,(x,-1,5),(y,-7,-2))

Parametrically (see also vector-parametric functions, parametric_plot)
Circle with center (0,0) and radius 3: r=vector( ( 3*cos(t),3*sin(t) ) )
var('t')
r=vector( ( 3*cos(t),3*sin(t) ) )
parametric_plot(r, (t,0, pi))
Notice that interval on t is [0,π]. Going from 0 to 2pi goes TWICE around the circle.

This works with off-center circles too except you will need to know how to create the vector-parametric function
Circle with center (2,-3) and radius \sqrt{5}: r=vector( ( sqrt(5)*cos(t)+2,sqrt(5)*sin(t)-3 ) )
To plot:
var ('x y')
r=vector( ( sqrt(5)*cos(t)+2,sqrt(5)*sin(t)-3 ) )
parametric_plot(r, (t,0, pi))

How to plot a circle in sage

A circle is defined either implicitly or parametrically. It is not defined explicitly.

Implicitly (see also implicit functions,implicit_plot)

Circle with center (0,0) and radius 3: x^2+y^2=9

To plot:

var ('x y')

implicit_plot(x^2+y^2==9,(x,-5,5),(y,-5,5))

Notice the "double" equals!

This works with off-center circles too except you will need to adjust your mins and maxes to get it.

Circle with center (2,-3) and radius \sqrt{5}: (x-2)^2+(y+3)^2=5

To plot:

var ('x y')

implicit_plot((x-2)^2+(y+3)^2==5,(x,-1,5),(y,-7,-2))

Parametrically (see also vector-parametric functions, parametric_plot)

Circle with center (0,0) and radius 3: r=vector( ( 3*cos(t),3*sin(t) ) )

var('t')

r=vector( ( 3*cos(t),3*sin(t) ) )

parametric_plot(r, (t,0, pi))

Notice that interval on t is [0,π]. Going from 0 to 2pi goes TWICE around the circle.

This works with off-center circles too except you will need to know how to create the vector-parametric function

Circle with center (2,-3) and radius \sqrt{5}: r=vector( ( sqrt(5)*cos(t)+2,sqrt(5)*sin(t)-3 ) )

To plot:

var ('x y')

r=vector( ( sqrt(5)*cos(t)+2,sqrt(5)*sin(t)-3 ) )

parametric_plot(r, (t,0, pi))

P.S. There is also a circle command if you just want to draw circles and not use the function in a calculation:

Reference: http://www.sagemath.org/doc/reference/sage/plot/circle.html

Formulas for advanced calculus: http://www.scribd.com/doc/77158858/3d-Integral-Formulas-Small-Eu