WOLFRAM|DEMONSTRATIONS PROJECT

Constructing Quadratic Curves

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parabola
ellipse
hyperbola
Drag the point A.
directrix
0.5
This Demonstration constructs the parabola, ellipse, and hyperbola geometrically. These constructions only need a straightedge and compass.
Here are the geometric definitions of these curves. A parabola is the set of points equidistant from a line (the directrix) and a point (the focus). A point
P
is on an ellipse if the sum of the distances from
P
to two other points (the foci)
F
1
and
F
2
is constant. A point
P
is on a hyperbola if the difference of the distances from
P
to two other points (the foci)
F
1
and
F
2
is constant; taking the difference one way gives one branch of the hyperbola and the other way gives the other branch.
Parabola: let
C
be the focus of the parabola, let
A
be a point on the directrix, and let
P
be the intersection of the perpendicular to the directrix at
A
and the bisector of the segment
AC
, so that
AP=CP
.
Ellipse: let
F
1
and
F
2
be the foci of an ellipse, let the point
A
be on the circle with center
F
2
and radius
r
. Let the point
P
be the intersection of the bisector of the segment
AF
1
and the straight line
AF
2
, so that
PF
1
+
PF
2
=r
.
Hyperbola: let
F
1
and
F
2
be the foci of a hyperbola, let the point
A
be on the circle with center
F
2
and radius
r
. Let the point
P
be the intersection of the bisector of the segment
AF
1
and the straight line through
A
and
F
2
, so that
PF
1
-
PF
2
=r
.
Line
BP
is always tangent to the curve at
P
.