In the
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s, an orthoptic is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of points for which two
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s of a given curve meet at a right angle.
Examples:
# The orthoptic of a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
is its directrix (proof: see
below),
# The orthoptic of an
ellipse is the
director circle
In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other. ...
(see
below),
# The orthoptic of a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
is the director circle
(in case of there are no orthogonal tangents, see
below),
# The orthoptic of an
astroid
In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it ...
is a
quadrifolium
The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation:
:r = a\cos(2\theta), \,
with corresponding algebraic equation
:(x^2+y^2)^3 = a^2(x^2-y^2)^2. \,
Rotated ...
with the polar equation
(see
below).
Generalizations:
# An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see
below).
# An isoptic of ''two'' plane curves is the set of points for which two tangents meet at a ''fixed angle''.
#
Thales' theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
on a
chord can be considered as the orthoptic of two circles which are degenerated to the two points and .
Orthoptic of a parabola
Any parabola can be transformed by a
rigid motion
Rigid or rigidity may refer to:
Mathematics and physics
*Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity
*Structural rigidity, a mathematical theory of the stiffness of ensembles of rig ...
(angles are not changed) into a parabola with equation
. The slope at a point of the parabola is
. Replacing gives the parametric representation of the parabola with the tangent slope as parameter:
The tangent has the equation
with the still unknown , which can be determined by inserting the coordinates of the parabola point. One gets
If a tangent contains the point , off the parabola, then the equation
:
holds, which has two solutions and corresponding to the two tangents passing . The free term of a reduced quadratic equation is always the product of its solutions. Hence, if the tangents meet at orthogonally, the following equations hold:
:
The last equation is equivalent to
:
which is the equation of the
directrix.
Orthoptic of an ellipse and hyperbola
Ellipse
Let
be the ellipse of consideration.
(1) The tangents to the ellipse
at the vertices and co-vertices intersect at the 4 points
, which lie on the desired orthoptic curve (the circle
).
(2) The tangent at a point
of the ellipse
has the equation
(see
tangent to an ellipse). If the point is not a vertex this equation can be solved for y:
Using the abbreviations
and the equation
one gets:
:
Hence
and the equation of a non vertical tangent is
:
Solving relations
for
and respecting
leads to the slope depending parametric representation of the ellipse:
:
(For another proof: see
Ellipse.)
If a tangent contains the point
, off the ellipse, then the equation
:
holds. Eliminating the square root leads to
:
which has two solutions
corresponding to the two tangents passing through
. The constant term of a monic quadratic equation is always the product of its solutions. Hence, if the tangents meet at
orthogonally, the following equations hold:
:
The last equation is equivalent to
:
From (1) and (2) one gets:
* The intersection points of orthogonal tangents are points of the circle
.
Hyperbola
The ellipse case can be adopted nearly exactly to the hyperbola case. The only changes to be made are to replace
with
and to restrict to . Therefore:
* The intersection points of orthogonal tangents are points of the circle
, where .
Orthoptic of an astroid
An astroid can be described by the parametric representation
:
.
From the condition
:
one recognizes the distance in parameter space at which an orthogonal tangent to appears. It turns out that the distance is independent of parameter , namely . The equations of the (orthogonal) tangents at the points and are respectively:
:
Their common point has coordinates:
:
This is simultaneously a parametric representation of the orthoptic.
Elimination of the parameter yields the implicit representation
:
Introducing the new parameter one gets
:
(The proof uses the
angle sum and difference identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
.) Hence we get the polar representation
:
of the orthoptic. Hence:
* The orthoptic of an astroid is a
quadrifolium
The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation:
:r = a\cos(2\theta), \,
with corresponding algebraic equation
:(x^2+y^2)^3 = a^2(x^2-y^2)^2. \,
Rotated ...
.
Isoptic of a parabola, an ellipse and a hyperbola
Below the isotopics for angles are listed. They are called -isoptics. For the proofs see
below.
Equations of the isoptics
; Parabola
:
The -isoptics of the parabola with equation are the branches of the hyperbola
:
The branches of the hyperbola provide the isoptics for the two angles and (see picture).
; Ellipse
:
The -isoptics of the ellipse with equation are the two parts of the degree-4 curve
:
(see picture).
; Hyperbola
:
The -isoptics of the hyperbola with the equation are the two parts of the degree-4 curve
:
Proofs
; Parabola
:
A parabola can be parametrized by the slope of its tangents :
:
The tangent with slope has the equation
:
The point is on the tangent if and only if
:
This means the slopes , of the two tangents containing fulfil the quadratic equation
:
If the tangents meet at angle or , the equation
:
must be fulfilled. Solving the quadratic equation for , and inserting , into the last equation, one gets
:
This is the equation of the hyperbola above. Its branches bear the two isoptics of the parabola for the two angles and .
; Ellipse
:
In the case of an ellipse one can adopt the idea for the orthoptic for the quadratic equation
:
Now, as in the case of a parabola, the quadratic equation has to be solved and the two solutions , must be inserted into the equation
:
Rearranging shows that the isoptics are parts of the degree-4 curve:
:
; Hyperbola
:
The solution for the case of a hyperbola can be adopted from the ellipse case by replacing with (as in the case of the orthoptics, see
above).
To visualize the isoptics, see
implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every impli ...
.
External links
''Special Plane Curves.''"Isoptic curve" at MathCurve"Orthoptic curve" at MathCurve
Notes
References
*
*
*
*
*
{{Differential transforms of plane curves
Curves