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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Planar Curves

[No one ignorant of geometry admitted]
Entrance of Plato's Academy in Athens, founded 387 BC

Related articles on this site:

Related Links (Outside this Site)

Famous Curves  in  MacTutor  (University of St. Andrews)
Two Dimensional Curves (2dcurves.com) by Jan Wassenaar
Classic Curves and Surfaces in the "National Curve Bank" of Gustavo Gordillo. Constructions with the Compass Alone

Planar Curves

Jessica (2008-08-19, Yahoo!)   Straight Line
What's the cartesian equation of the line through  (-8,9)  and  (10,-3) ?

For a point (x,y) on that line, the vector from (-8,9) to (x,y) is proportional to the vector from (-8,9) to (10,-3).  The relevant determinant vanishes:

|     10 + 8
-3 - 9
      x + 8
y - 9
    |   =   0

Thus, 18(y-9) + 12(x+8) = 0.  which boils down to the equation of the line:

2 x  +  3 y   =   11

 Confocal Conic Sections
(2004-06-23)   Confocal Conics
Ellipses and hyperbolae sharing the same foci.

Ivory's Theorem:  In a curvilinear quadrangle formed by two pairs of confocal conics, the diagonals are equal.

 Come back later, we're
 still working on this one...

The conic sections were named by  Apollonius of Perga (262-190 BC).
Dandelin spheres (1822)  are due to  (Pierre) Germinal Dandelin (1794-1847; X1813)
partial credit also goes to  Adolphe Quételet (1796-1874)
Hamilton (1805-1865)  remarked that a Dandelin sphere touches the cone in a plane containing a directrix.
In 1829, Pierce Morton (1803-1859) used Dandelin spheres to prove the focus-directrix property of conics.

 Spiral of Archimedes Jerry Southerland  (2005-02-02; e-mail)
Spiral of Archimedes
What's the length of the spiral from a vinyl record (or a paper towel roll) as a function of the distance to the center?

spiral of Archimedes  may be described as the trajectory of a point which moves at constant speed along a spoke of a wheel rotating at a constant rate.

As the distance (r) from the center to a point on the spiral is proportional to the angle (q) between the spoke and some fixed line,  the polar equation is:

r   =   a q

The angle q (expressed in radians) increases by 2p with each turn.  Therefore, the quantity  2pa  is the thickness of the paper on a roll,  or the distance between adjacent parts of  the  groove on a vinyl record.

When the polar coordinates  (r,q)  of a point change minutely by  (dr,dq),  that point travels a minute distance ds, given by the following relation (obtained from the Pythagorean theorem, since a smooth curve  looks straight  at small enough scales):

(ds)2   =   (dr)2 + (r dq)2

In the case of Archimedes' spiral,  this means:   ds  =  a (1 + q 2 )½ dq.  The length (s) of the spiral is obtained by integrating this:

vinculum vinculum
s    =     ½ aq  Ö 1 + q2   +  ½ a ln (  Ö 1 + q2  + q )
vinculum vinculum
= ½   r  Ö 1 + (r/a)2   +  ½ a ln (  Ö 1 + (r/a)2  +  r/a )

For a roll of paper of diameter 2r1 with a core of diameter 2r0 the total length is simply   s(r1)-s(r0).  [Recal that  2pa is the thickness of the paper.]

A Simple Approximation:

The  exact  formula above is an  overkill  for estimating the length of paper on a roll.  Instead, you may consider that, if the paper is very thin compared to the diameter of the core, the surface area of the roll's cross section divided by the thickness (e = 2pa) of the paper gives you its length, namely:

s     »     (r12 - r02 ) / 2a     =     p (r12 - r02 ) / e

This approximation is just the leading term of the above exact formula.

On 2005-02-11, Jerry Southerland wrote:     [edited summary]
For a 65 ft roll of paper towels,  2r1 = 5"  and  2r0 = 1.75".
The thickness of the paper, measured with a caliper, is 0.01".  Plugging this into the formula yields a length of about 144 ft, which is more than twice the actual value.  What's wrong?

A paper towel normally has features  (fluffiness, waviness, embossment)  which make the caliper measurement of a single sheet greatly underestimate the effective thickness of the material in a stack or a roll.  By working the formula backwards, the above numbers would indicate that the paper material has an effective thickness of about 0.022".  This could be obtained directly by counting the number N of turns it takes for the diameter of the roll to decrease by x inches  (the effective thickness of the stuff would then be   x / 2N).  If you insist on using a caliper, you may reduce by a factor of N the large error obtained for a single sheet, simply by using the caliper (without squeezing) on a stack of N sheets...

If a roll of paper towels of the type quoted above has a diameter D (in inches) then its footage is approximately:    ( D 2 - 3.06 ) 2.96 ft

(2009-12-06)   Hyperbolic Spiral   (Pierre Varignon, 1704)
Also called  reciprocal spiral.  Inverse of the  Archimedean spiral.

r   =   a / q

MacTutor   |   MathWorld   |   Wikipedia

(2008-03-29)   Catenary  or  chainette
The shape of a  chain  under its own weight :    y + a   =   a  ch(x/a)

A thin uniform rope without any rigidity is traditionally called a  chain  (in Latin:  catena).  The lack of any force sideways means  there's just a  tangential  tension.  Let's characterize the shape  (catenary)  of a chain under its own weight:

Let  j  be the  inclination  of the chain  (i.e., the angle its tangent makes with the horizontal x-axis)  at a given curvilinear abscissa  s.  The tension pulling on the left part of the chain has magnitude  F  and coordinates  (F cos j, F sin j).  For a positive displacement  ds  along the rope, the infinitesimal increase of the vertical component is balanced by the weight of the corresponding piece of rope  (we assume the  mass per unit of length  to be a constant  m ) :

d ( F sin j )   =   g m ds

As the horizontal component of the tension is a constant:  F = K / cos j

d ( K tan j )   =   g m ds

A Cartesian relation is readily obtained by using  y'  = dy/dx = tan j

K d(y' )   =   g m ( 1 + (y' ) 2 ) ½  dx

This is integrated as a simple quadrature, by considering  x  as function of  y' :

x   =   xo  +  a  Argsh ( y' )       [ introducing  a = K/gm ]

Conversely,  y'   =   sh [(x-xo )/a].  Therefore, by integration:

y   =   yo  +  a  ch [(x-xo )/a]

Thus, if we use the apex of the curve as origin of the coordinates, we have:

y   =   a  ch (x/a)  -  a   =   2 a  sh2 (x/2a)

The  curvilinear abscissa  is simply :   s   =   a  sh (x/a)   =   a  tan j

For a plane curve, such a relation between the  arclength  s  and the  inclination  j  goes by the name of  Whewell equation.  It can be used to derive the so-called  intrinsic Cesàro equation  which relates the (signed) radius of curvature  R = ds/dj  and the arclength  s.  In the case of the catenary, we have:

R   =   ds/dj   =   a  ( 1 + tan2 j )   =   a + s2/a

So, the parameter  a  is the radius of curvature at the catenary's apex  (s=0).

Galileo (1564-1642) thought that a chain under its own weight would take on a parabolic shape,  which Jungius (1587-1657) disproved.  The latin name  catenaria  was coined by Huygens in 1690  (in a letter to Leibniz).  This was immediatly translayed as  "chainette"  in French.  In English, the locution "catenarian curve" was used at first.   Arms of Thomas Jefferson Thomas Jefferson introduced the word  catenary  a century later  (1788).  The above equations for a catenary were first derived in 1691, by Leibniz, Huygens and Johann Bernoulli,  in their respective responses to an open challenge from  Jakob Bernoulli.

The Catenary - Mathematics all around us,  by James Grime  (singingbanana).   |   Wikipedia :  Catenary

(2015-02-15)   Tractrix   (Huygens, 1692)
It's at a constant distance  a  from its asymptote along the tangent.

The  evolute  of the  tractrix  is the  catenary.

The surface generated by revolving the tractrix about its asymptote is known as the  tractricoid.  It has a finite surface area of  4pa 2  and it encloses a finite volume of  2/pa 3.

The  tratricoid  is a surface of  constant  negative  Gaussian curvature.  Such a surface is known as a  pseudosphere  and provides a concrete model for the  hyperbolic geometry  first described axiomatically  (negating Euclid's fifth postulate)  by Lobachevsky and Bolay  (Beltrami, 1868).

MacTutor   |   Wikipedia :  Tractrix   |   Tractricoid

(2003-05-25)       Why is it called a "witch" ? Witch of Agnesi (versiera)
The witch of Agnesi.
Agnesi's cubic  (versiera).
[French: agnésienne]
[Spanish: bruja de Agnesi]

The curve's cartesian equation is:  y = a3 / (x2+a2 ).  It's associated with a circle of diameter a tangent to the x-axis at the origin O.  A point M on this curve may be obtained via a line OPQ where P is on the circle and Q is at height a; M has the x-coordinate of Q and the y-coordinate of P.

This curve was first considered by:  Maria Gaetana Agnesi 
 (1718-1799)  Pierre de Fermat 

It seems you've got to be about 30 years of age to be interested in this...   Just a joke!

The Italian name versiera (used by Agnesi) was actually given to the curve by Grandi in 1718, because of the "inverted sine" (sinus versus) which is used in its construction.  In spite of what is often stated, there may not be any direct relation with the Latin word versoria which denotes the rope used on a sailboat to transfer edge (the verb vertere means "to turn").  The weird name of "witch" comes from a famous mistranslation by the Lucasian professor of mathematics at Cambridge, John Colson (1680-1760).  When he translated Maria Agnesi's Istituzioni Analitiche  [written in 1748, this was the first calculus book ever authored by a woman]  Colson apparently mistook "la versiera" for "l'avversaria"...  Since l'avversario (the adversary) is the Devil, an avversaria would be a she-devil, or a witch!  Colson's work seems also responsible for the attribution of the curve to Agnesi (she made no such claim herself).

The area between the versiera and its asymptote  [the x-axis]  is  pa2,  which happens to be exactly 4 times the area of the associated circle.

The function   a / p(x2+a2 )   thus corresponds to the density of a continuous probability distribution  (whose  standard deviation  is  infinite ).

Folium of Descartes  

(2004-06-23)   Folium of Descartes  (1638)

Its cartesian equation is:     x 3 + y 3  =  3axy

The asymptote is of equation:     x + y + a  =  0
The surface area of the lobe is  3 a 2/ 2  .  So is the area between the asymptote and the curve.

The total signed area bounded by the  oriented  closed loop consisting of the folium and its asymptote is  zero  (that loop is understood to include 2 points "at infinity" and to have a continuous  tangent vector  at the double point).

Lemniscate of Bernoulli
(2004-06-21)   Lemniscate of Bernoulli

 Jakob Bernoulli 

The Latin word "lemniscus" means pendant ribbon.  This name was first used in 1694 by Jacob Bernoulli (1655-1705) to describe the above planar curve, which is now named after him.

lemniscate of Bernoulli  is the particular Cassinian oval which goes through the midpoint between its  foci.  If  2a  is the distance between the foci, the lemniscate's cartesian equation in the proper coordinate system is:

( x 2 + y 2 ) 2   =   2 a 2 ( x 2 - y 2 )

The horizontal width of the curve is 2Öa, while its height (or vertical width) is simply a.  The surface area of each lobe happens to be simply a2.

(2004-06-21)   Ovals of Cassini   /   Cassinian Ovals   /   Cassini Ovals
 Cassini Ovals  
 Jean-Dominique Cassini 
Cassini ovals are planar curves along which the product of the distances to two given points, called foci, is a constant [ foci, plural of focus, is pronounced "foe-sigh"].  By contrast, the sum of these two distances is a constant along an ellipse...

Generalized Definition of a Cassini Ovum :

A multipolar  Cassini ovum  can be defined as the set of all points whose distances to a certain number of fixed  poles  have a constant product.

In the plane, such sets are curves  (circles in the monopolar case, traditional ovals of Cassini in the dipolar case).  In 3-dimensional Euclidean space, they are 2-dimensional surfaces  (possessing cylindrical symmetry in the dipolar case only).  A Cassini ovum in higher dimensions is an hypersurface.

The above generalized definition isn't universally accepted.  It's prudent to restate it before use, possibly with a link to this note:


(2004-06-22)   Cardioid & Limaçons of Pascal

 Blaise Pascal (1625-1712) 
 son of Etienne Pascal  

These curves are named after Etienne Pascal (1588-1651), the father of the French scientist and philosopher Blaise Pascal (1623-1662).  Etienne Pascal was part of a circle of mathematicians who held weekly meeting in Paris, around Marin Mersenne (1588-1648).

Limacons of Pascal This famous mathematical group included the likes of Pierre Hérigone (1580-1643), Claude Mydorge (1585-1647), Girard Desargues (1591-1661), Pierre Gassendi (1592-1655), Jean Beaugrand (1595-1640), René Descartes (1596-1650), Claude Hardy (1598-1678), Pierre de Carcavi (1600-1684) and also Gilles Personne de Roberval (1602-1675)  who is the one who actually named the limaçon after Etienne Pascal in 1650, when he used it as an example for the drawing of tangents.

A limaçon can probably be defined in more different ways than any other quartic curve.  It can be described in any of the following equivalent ways:

  • Curve of polar equation   r = a cos(q) + b
  • Curve of cartesian equation   ( x 2 + y 2 - ax ) 2  =  b 2 (x 2 + y 2 ) 
  • Epitrochoid of unit ratio, namely:  The trajectory of a point at a distance ½ a  from the center of a circle of diameter b which rolls on a fixed circle of the same size.  This is called a unit epicycloid if  a = b  (cardioid). 
  • Conchoid of a circle (of diameter a) with respect to one of its points.  In general, the conchoid of a curve of polar equation   rf (q)   is the curve of polar equation   rf (q) ± b.   Here,   ra cos(q)   is indeed the polar equation of a circle of diameter a which goes through the origin. 
  • Pedal curve of a circle of radius b with respect to a pole at a distance a from its center.  In general, the pedal of a curve is the locus of the foot of the perpendicular from the pole to a tangent of the curve. 
  • Inverse of a conical section with respect to one if its foci. 
  • Cartesian oval containing one of its own poles. 
  • Envelope of the circles centered on a fixed circle and containing a given point. 
  • Etc.

cardioid is a limaçon for which  a = b.  A circle is obtained for b = 0.

(2004-06-22)   Cartesian Ovals   /   Ovum of Descartes


Given two poles A and B, a  cartesian oval  is the set of all points M for which   AM + l BM   is a constant  (for some fixed parameter l).

Ovals of Descartes

This three-parameter family of curves includes the above [two-parameter] limaçons of Pascal as a special case (corresponding to the cartesian ovals which go through a pole).

 Come back later, we're
 still working on this one...

(2012-11-18)   Envelope of a parametrized family of curves
At every point, the envelope is tangent to a curve from the family.

If the cartesian equation of each curve is  F(x,y,t) = 0  for a fixed value of the continuous parameter t, then the equation of the envelope is obtained by eliminating t between the two equations:

F = 0     and     F / t = 0

 Come back later, we're
 still working on this one...

Envelope of a family of curves

(2012-11-20)   Evolute of a smooth curve   (French:  développée )
The  evolute  is the envelope of all the  normals  to the curve.

 Christiaan Huygens
 (1629-1695) A concept due to Huygens (1673).  The tangents of the evolute are normal to the curve.  A smooth curve is the evolute of any involute.

The  evolute  is also the locus of the curve's centers of curvature.

Evolute   |   Christiaan Huygens (1673)

(2012-11-20)   Involute of a smooth curve   (French:  développante )
Trajectory of a point on a line that rolls  (without sliding)  on the curve.

 Christiaan Huygens
 (1629-1695) A concept due to Huygens (1673).  The normals of the involute are tangent to the curve.  A smooth curve is an involute of its evolute.

The involute can also be envisioned as the trajectory of the extremity of a taut string that is  wound  onto  (or  unwound  from)  the curve.

If you must, use this latter viewpoint piecewise for parts of the curve whose curvature has a constant sign.  There's no mathematical difficulty when the curve crosses its tangent, but then it can't really be viewed as the perimeter of an actual spool of string.

Of particular interest is the  involute of a circle  (which is what the term  "involute" is understood to denote when it doesn't come with a qualifier).  This is the  only  tooth profile  that makes a driven  spur gear  rotate at a uniform rate  (or "steady speed")  when the driver does.  All other types of spur gear will thus induce systematic vibration that are unacceptable for high-speed heavy machinery  (but perfectly fine for low-speed clockwork in which pulsed rotation is the norm).

Involute   |   Christiaan Huygens (1673)

 (1646-1716) (2009-02-21)   Parallel Curves   (Leibniz, 1692)
Two curves which stay at a constant distance  (locally).

The envelope of the circles of radius  a  centered on a curve  G is generated by the two points at distance  a  on the normal to G at a running point  P.  For a simple closed curve, this envelope can be divided into two separate curves; the  inner  and  outter  parallel at offset  a, which are respectively generated by the points on the normal which lay on the inward or outward side of the normal.  (In the field of computer graphics, parallel cfrom each other by translation.

The perimeters of a smooth closed curve and that of one of its  [single-sided]  parallels at offset  a  have either a sum or a difference equal to  2pa.

(2009-11-22)   Nephroid,  two-cusped epicycloid  (1678)
Point on the circumference of a disk rolling outside a circle twice as big.

This curve was introduced in 1678 as the  catacaustic  of a circle by  Christiaan Huyghens (1629-1695) who would later discuss it in  Traité de la lumière  (1690).

The term  nephroid  was coined by the English astronomer Richard A. Proctor (1837-1888) in  The Geometry of Cycloids  (London, 1878).

x   =   a ( 3 cos t  -  cos 3t )
y   =   a ( 3  sin t  -   sin 3t )

David Darling   |   MacTutor   |   MathWorld

(2009-11-22)   Freeth's Nephroid,  Nephroid of Freeth  (1879)
Strophoid of a circle (pole at center & fixed point on circumference).

Freeth's nephroid  was first described  (with other strophoids)  by the vicar Thomas Jacob Freeth (1819-1904) in 1879  (London Mathematical Society).

r   =   a  ( 1  +  2 sin q/2 )

MacTutor   |   MathWorld

(2005-06-01)   Bézier Curves
Standard parametric curves used to approximate other curves.

Bézier curves are often restricted to planar geometry, but they need not be.

Pierre Bézier (1910-1999) served as an engineer for the French car manufacturer Renault for 42 years, from 1933 to 1975.  He was also a professor at the  Conservatoire national des arts et métiers  (CNAM)  from 1968 to 1979.  He introduced the type of curves now named after him for "free-form curves" within the CAD/CAM system dubbed UNISURF, which he launched in 1968.

The order-n  Bézier curve from P0 to Pn involves  n+1  control points  (P).  A point  M(t)  of the curve is the following barycenter of the control points:

M(t)   =    
i = 0
  t i (1-t) n-i  Pi

Unless otherwise specified, order  n = 3  is understood, in which case alternate names may be used:  "Bézier cubic", "cubic spline" (or just "spline").  Such a curve is given by its extremities A and B and two other control points P and Q:

M(t)   =   (1-t) 3 A   +   3 t (1-t) 2 P   +   3 t 2 (1-t) Q   +   t 3 B

 Bezier curve

As t goes from 0 to 1, M(t) goes from A to B.  Assuming distinct control points, the line AP is tangent to the spline at point A, whereas BQ is the tangent at point B.

It's good to know that if PQ is parallel to AB and  PQ = AB/3,  then the Bézier cubic spline is actually a  parabolic arc.  Thus, it's easy to obtain parabolas from any computer graphic package that provides Bézier cubics  (most of them do).  Bézier curves of order 2 (with a single intermediary control point R)  Quadratic
 Bezier Curve correspond to these same  parabolic arcs, since the relations
P = (A+2R)/3   and   Q = (2R+B)/3   imply:

 (1-t) 3 A  +  3 t (1-t) 2 P  +  3 t 2 (1-t) Q  +  t 3 B
=     (1-t) 2 A   +   2 t (1-t) R   +   t 2 B

Any Bézier curve of order n  (with control points  P) is also a Bézier curve of order n+1 with control points  Q given by the following relation.  Note that:  Q0 = P0  and  Qn+1 = P   [HINT:  Expand  (1-t) M(t) + t M(t) ]

 n +1 
 Qi    =     æ
 i -
 Pi-1  +   æ

Pascal's formula  makes this a barycentric relation:  Qi  is between  Pi-1 and Pi  (e.g., in the above parabolic-to-cubic case, P is on AR and Q is on RB).

Dr. Nicholas A. Sceusa (2005-05-29; e-mail)   Piecewise Circular Curves
How may an approximation to an arbitrary curve be constructed
using compass and straightedge?

In the computer era, the most popular approximations for arbitrary curves involve algebraic splines, like the Bézier curves discussed above.  For more traditional work, the circle remains, indeed, the basic curve of choice.

The following procedure assumes that the curve we want to approximate contains no straight sections and no angular points  (otherwise we'd process separately each piece of the curve between these).

To build a smooth curve consisting of circular arcs which approximates the original curve, we first choose a number of "construction points" located on that curve.  If the curve is open, we must include its extremities among these.  It's also a good idea to include any  inflection points  the original curve may have.  (Inflection points  are points where the curve crosses its own tangent.  Near one of its inflection points, the curve is usually very close to a straight line.)

  • Draw the tangent to the curve at each construction point.
  • Join two consecutive construction points by a circular arc centered at the point where the perpendiculars to the relevant tangents intersect.

Such lines always intersect if all inflections points have been included among the construction points; parallelism would indicate that we're bracketing an inflection point and need more intermediary construction points.  (A practical alternative is to "cheat" by introducing small straight sections around inflection points.)

The intrinsic equation of a piecewise circular function is a step function...

(2005-06-06)   Planar Shape:  Intrinsic Equation of a Planar Curve
Defining a shape by its curvature, as a function of curvilinear abcissa.

The curvilinear abscissa (often denoted s) is the arc length  measured along the curve from some arbitrary origin on it  (counted negatively "before" this origin).

At a particular point, the  curvature  of a planar curve has a magnitude equal to the reciprocal of its radius of curvature  (the radius of the so-called  osculating circle  which best approximates the curve in the immediate neighborhood).  By convention, the  sign of the curvature is  positive  if the curves turns to the  left of the direction of an increasing curvilinear abscissa.  The curvature is zero at an inflection point, or at any point of a straight section.

The so-called  intrinsic equation  (which gives the curvature of the curve as a function of the curvilinear abscissa along it)  uniquely  defines the "absolute" shape of a planar curve  (it could be translated or rotated, but not flipped over).

Infinite curvature: "Planar shapes" are distributions

The above viewpoint is only adequate for smooth curves with no angular points.  However, it's easy to generalize it to angular curves, since the curvature is the derivative [with respect to curvilinear abscissa] of the angle (in radians) between some fixed direction and the tangent to the curve.  A sharp turn of  q  radians at a point so of infinite curvature thus translates into a Dirac  spike  q d(s-so)  in the  intrinsic equation  (namely, a Delta distribution of amplitude q ).

Dr. Nicholas A. Sceusa  (2005-05-31; postal mail)
On the Quadratrix (or Trisectrix) of Hippias
Could  Hippias  draw his quadratrix with straightedge and compass?

The quadratrix is the trajectory of the intersection of two straight lines:  One in uniform motion, the other uniformly rotating about a fixed point A.

 Quadratrix of Hippias

Infinitely many constructible angles can be used to obtain points of the quadratrix.  In particular, as any angle can be bissected with straightedge and compass, we may divide a circle in 2, 4, 8, 16, 32, 64... 2 n equal angles.  Draw such radial lines for a sufficiently large n, then draw any set of equally-spaced parallel lines.  The points where successive radial lines intersect successive parallel lines are on a quadratrix...  (Intermediary points may be obtained by doubling up the lines in both sets.)

The ancient Greeks only considered those points which could be deduced from a few given points in finitely many uses of a straightedge or a compass.  The quadratrix of Hippias was the first curve on record whose points cannot  all  be so deduced.  Thus, it's not considered a   constructible  curve, although infinitely many points on it can be constructed classically, as shown above.

In the proper coordinate system, the cartesian equation of the quadratrix is:

y   =    a -  x / tan(x/a)

The center of rotation (A) is at  (x,y) = (0,a).  The curve's apex (O) is at  (0,0).  The main branch is bounded by two vertical asymptotes (-p < x/a < p) and corresponds to a full turn of the rotating line.  Only the central portion is usually considered (-p/2 < x/a < p/2) which corresponds to a half-turn of the rotating line, between horizontal positions.  The portion where x is positive may suffice.

The curve was devised around 420 BC  by Hippias of Elis.  It's been called either  quadratrix  or  trisectrix  because it provides graphic solutions to two famous classical problems, which are now known to be  outside the scope  of what can be done with straightedge and compass:  The squaring (quadrature) of the circle and the  trisection  of an arbitrary angle.

Also, Heath has attributed to Archytas of Tarentum a "quadratrix solution" to the third most famous classical problem of this kind, the  duplication of the cube, or Delian problem.  Legend has it that the Delian problem dates back to 430 BC, when the Oracle of Apollo at Delos said the plague ravaging Athens would come to an end if the altar of Appollo could be made  exactly  "twice as large" [by volume].  The cube root of 2 is still occasionally known as the  Delian constant.

The quadratrix can be used to "square the circle" because the number p (which can't be constructed with straightedge and compass) is simply obtained as the ratio of the width to the height of the section of the quadrarix whose extremities are on an horizontal line through A.

To obtain a portion l of an arbitrary angle (including l = 1/3) whose vertex is at point A, consider the two points where its sides intersect the main branch of the quadratrix.  Draw trough them 2 lines parallel to the axis of the quadratrix.  If M is the point where the quadratrix intersects a third parallel line dividing the space between them in the ratio l, then the line AM divides our angle in the ratio l.

This wonderful curve is thus directly related to yet another famous problem:  the construction of an n-sided regular polygon.  Equilateral triangles, squares, regular pentagons and hexagons are all  constructible  with straightedge and compass, but it turns out that regular heptagons, enneagons, hendecagons, or tridecagons  cannot  be so obtained.  This was established in 1796 by Carl Gauss (at age 19) who showed the heptadecagon (17 sides)  to be classically  constructible.

(2005-06-06)   A Parabola is an Example of a Constructible Curve
Locating any point on a parabola, with only straightedge and compass.

A parabola is the set of all points of the plane at an equal distance from a given point (called the parabola's  focus) and a given line (the parabola's  directrix).

The parabola is called "constructible" because its intersection with an  arbitrary  straight line, or an  arbitrary  circle, can always be located in finitely many steps using only straightedge and compass.  Thus, allowing the use of of a parabolic spline in addition to straightedge and compass would not introduce any additional "constructible" points.  This could make classical geometry easier, but not richer.

There is only One True Parabola  by  Matt Parker   (2016-03-03).

(2005-06-20)   The Mohr-Mascheroni Theorem:
All constructible points can be obtained with the compass alone.
 Compass   This statement about classical planar geometry used to be credited to Lorenzo Mascheroni (1750-1800) who proved it in  Geometria del compasso  (1797).  However, 125 years earlier, the Danish mathematician Georg Mohr (1640-1697) had published anonymously a book entitled Euclides Danicus (1672) which contains a prior proof...  This was completely overlooked until 1928, when a copy of that little-known work surfaced in a bookstore  (there's no record of any copy ever sold before then).  Classical constructions which avoid the use of a straightedge are thus now best called  Mohr-Mascheroni constructions.
Georg Mohr had also shown that all Euclidean constructions could be done with only a straightedge, if a circle of given center is already drawn somewhere in the plane.  The Swiss mathematician Jakob Steiner (1796-1863)  rediscovered this in 1833  (Steiner constructions).

We may prove that all constructible points can be obtained with the compass alone by providing Mohr-Mascheroni constructions for the two types of points which normally involve the use of the straightedge:

  • The intersections of a straight line and a circle.
  • The intersection of two straight lines.

We'll do so using only  two  explicit Mohr-Mascheroni constructions, concerning the inversion with respect to a circle and the circumcenter of three points.  The very simple construction we'll provide for the former isn't completely general unless supplemented by a scaling-up procedure, like the following one:

Doubling up, with the compass alone :

 Doubling a segment, 
 with compass alone.

An example of a Mohr-Mascheroni construction is the  doubling, which we'll use below:  Given two points O and M, the point N in the direction of OM such that ON = 2OM may be obtained as follows:

Draw the  entire  circle (C) of center M and radius OM.  It intersects the circle of center O and radius OM at points A and B.  The circle of center A through B (or center B through A) intersects (C) at point N  Halmos

(It wouldn't be acceptable to define N as the intersection of the circles centered on A and B, because there are  two  such points of intersection, both "new".)

Inversion of the plane, with respect to a circle :

The so-called  inversion  of the Euclidean plane with respect to a circle of center O and radius R  is a transformation which was introduced in 1826 by Steiner  (in his first published paper).  The  inverse  of a point M (other than O) is the point N in the direction of OM which is such that  OM.ON = R2.  This transforms a circle through O into a straight line, and vice-versa.  Circles not containing O are simply associated with other such circles.


N is very simple to obtain from M with the compass alone:  Assume OM is more than R/2 and consider the intersections A and B of the invariant circle with the circle of center M and radius OM.  Let N be the intersection (besides O) of the circles of centers A and B going through O.  We have:

OM . ON   =   R 2

This relation is easy to prove with cartesian coordinates:  O=(0,0), A=(x,y), N=(2x,0), M=(a,0).  Since, R=OA and MO=MA,  we have:  R2 = x2 + y2 and a2 = (a-x)2 + y2.  By subtraction:  R2 = 2xa = OM.ON   Halmos

If OM is R/2 or less, we may first obtain a point M' such that  OM' = 2n OM, using the above doubling n times, for a value of n large enough to make OM' greater than R/2.  The inverse N' of M' is then constructed as above.  Finally, the inverse N of M is simply obtained by doubling ON' n times.

Circumcenter of 3 given points :

The construction, with straightedge and compass of the center of the circle circumscribed to a triangle is a very early piece of mathematics; the circumcenter of a triangle is the intersection of the perpendicular bissectors of two sides.  The following Mohr-Mascheroni construction uses an inversion centered on one of the 3 vertices to obtain the inverse of the circumcenter as the intersection of the two circles which are inverses of these bissectors.

Consider a triangle ABC, whose circumcenter N is sought.  In an inversion of center A and radius AB, the image of the perpendicular bissector of AB is a circle U of center B and radius AB.  On the other hand, the inverse  [with respect to the circle U] of the perpendicular bissector of AC is a circle V of diameter AH', where H' the inverse of the middle H of AC.  The center of V is thus the inverse C' of C (constructed as indicated above).  The circles (U and V) through O of centers B and C' intersect at another point M, whose inverse (with respect to U) is the circumcenter N of ABC   Halmos

Of course, once the circumcenter is thus obtained, with compass alone, the entire circle through A, B and C is easily drawn...

Doing away with the straightedge entirely:

In classical geometry, any planar figure consists of straight lines and circles:

  • A straight line is given by two of its points. 
  • A circle is given by its center and a point on its circumference.
    It may also be defined by 3 points on its circumference  (Euphorbus).

Consider an inversion of radius R whose center O is  distinct  from the finitely many points of intersection of such a classical figure (it's best to let the distance of O to all such points be between R/2 and R so the simplest compass construction can be used to go back and forth between a point and its image):

  • The inverse of the straight line trough the points A and B is the circle going through O and the respective inverses A' and B' of A and B. 
  • The inverse of the circle through A, B and C is the circle going through their inverses A', B' and C'.

These remarks and the above explicit compass-only constructions allow the inverse of all the elements of a Euclidean planar figure to be drawn with compass alone.  The intersections of such elements can be constructed with compass alone as the inverses of the intersections in the inversed figure.   Halmos

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