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conic_eccentricity.gc

Focus and Directrix of Conic Sections

See the connection among ellipses, parabolas, and hyperbolas by changing
how a vertex sits relative to a focus and a directrix..

1. Drag the red dot on the vertex of the ellipse until
the ellipse becomes a parabola (pictured in grey).
2. Drag the vertex further and the parabola becomes a hyperbola.
Keep dragging the vertex down until you see both its arms.
3. Try dragging the red dot on the directrix to move the entire directrix.
What happens?
4. Play!

Bruce Simmons
Austin, TX
December 9, 2001

p=0.125+0.109375*i

q=-1.375-(0.359375*i)

h=Re(p),k=([Im(p)]^2+[Im(q)]^2)/(2*Im(p))

b=k-Im(p),a=sqrt(b^2-[k+Im(q)]^2),d=sqrt([k+Im(q)]^2-b^2)

y=Im(q)

[x-Re(p)]^2+[y+Im(q)]^2<0.001

[x-Re(p)]^2+[y+Im(q)]^2=0.001

vector(x,y)=vector(a*cos(2*pi*t)+h,b*sin(2*pi*t)+k)

y=-1/(4*Im(q))*[x-h]^2

vector(x,y)=vector(d*sinh([10*t-5])+h,-(b*cosh([10*t-5]))+k)

vector(x,y)=vector(d*sinh([10*t-5])+h,b*cosh([10*t-5])+k)

Graph of the formula

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