5 Point Ellipse

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The Five Point Ellipse

Marco Carpiceci - Department of Drawing and Survey,

University of Study in Rome - La Sapienza

[email protected]

Excerpted from an article originally published in"disegnare / idee immagini" Anno VII, n. 13, December

1996,ISBN 88-7448-739-8.(Note - paragraph and illustration numbering conforms to the original article).

Revised and expanded by Peter A. Luft([email protected]), with translation support from GianLuca Sabbi ([email protected]). Technical terms

have been changed to American usage. Supplemental illustrations 15b, 16b, 24-26 have been added, and

existing illustrations have been enhanced with color and additional labels for clarity. Paragraphs 15b and

16b have been added. This document last updated on 04/23/2009 by Peter A. Luft.

14) A conic (section, e.g. circle, ellipse, parabola, hyperbola ) is completely determined

by five points or five tangent lines. Because of the nature of conics, its possible todetermine the sequence of points defining the locus of the curve [Note 1], or the sequenceof tangents defining the envelope of the curve [Note 2].

15) The Theorem of Pascal [Note 3] and itsdual [Note 4] define exact conditions that

connect six points or six lines of a conic.

However, we only need five to define a

conic. The sixth point or line is redundant,and can therefore be defined as required,

depending on what the goal is. So if the goal

is, for example, determining a chord [Note 5]or a tangent [Note 6], you can define a sixth

point accordingly.

16) By choosing, for example, an orientation

of a new chord parallel to two given vertices,

you can thus obtain the length of the chord,

and as a consequence you obtain an axis ofoblique symmetry which passes through the

midpoint of a pair of parallel chords, and the

direction of its compliment [Note 7]. If you

repeat the procedure for a second time,you will find a second (independent)

conjugate diameter and the direction of its

complement. Taking both pairs together,you obtain the center of the conic curve.

This center is ideal (at infinity) in the

case of the parabola, but is ordinary

(finite) for the hyperbola and ellipse seeFig. 5.

In accordance with the involutionaryproperties of conjugate diameters,you can

trace any circle with center W, thatpasses through the center of the coniccurve, and then connect the intersections

of the two conjugate diameters. These

straight lines will all meet in a point V,

called the center of involution. This center

will be either external or internal to the


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circle, depending on whether the conic curve is an ellipse or a hyperbola (if the conic is a

circle, Vcoincides with W). The involutionary line, that is, the line connecting V with W,

will locate points X and Yon . The lines joining these points with Oare the x and yaxes of the ellipse, its unique pair of orthogonal conjugate diameters. See Fig. 6.

17) Using the projective characteristics of orthogonal affinity between the ellipses and thetwo limiting circles, inscribed and circumscribed, Ive developed the following graphic

method. It hasnt been described in any treatise I've ever seen, and is based on knowingthe axes and two general points of the conic.

After determining the position of theorthogonal x and y axes as described

above, take two points A and B on the

ellipse, and trace a straight line hcontaining these two points. The

intersection of this line with the x and y

axes yields the pointsHyandHx, as shownin Fig. 7. Then project the pointsA,BandH- the mid-point between AandB- onto

the Yaxis, and trace the arc, the diameter

of which is determined by the vertices Oand Hy. Point H' is now the point of

intersection of this circumference and the

ray projectingHontoy.

18) Now trace the line h'passing through

HyandH'. The intersection of h' with the

two rays projectingAandBontoyoccursat the points A' and B', and through thesepoints will pass the circle ry of center O

and the radius OA'. In fact, the straight

line passing through H' and O is

perpendicular to h' at the mid-point ofA'B', and so bisects the angle A' OB'. The

circumference ry is determined by the

maximum radius of the ellipse relative tothe yaxis, and will determine the vertices

of the ellipse on theyaxes.

19) Similarly, we can also project ABand

Honto the xaxis as shown in Fig. 8, and

trace the circumference of diameter OHx.H" is the point of intersection of the

circumference with the ray projecting from

Honto the x axis. Then trace the straight

line h" through Hx and H". Theintersections between h"and the two rays

projecting through A and B through x

determine points A" and B" of thecircumference rx, and this circumference

will determine the vertices of the ellipse

on thexaxis.

20) At this point, if we have an elliptical

compass all wed need to do is position it

correctly on the axes, set the minimum and maximum limiting radii, and trace the curve.Otherwise, if we have any CAD software available, we would trace the conic by

indicating the center and the two vertices of the ellipse on the two axes.

21) In case you had to trace the ellipses using only points because you didnt have any

instruments, but youd already determined the center and the two pairs of orthogonal


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vertices, one of the most widely used methods available would that of transformed points

or limiting circles, also called the method of double orthogonal affinity[Note 8].

The circle in projection

22) The graphics problem of the conic curves are usually treated within the disciplines ofrendering existing at different levels in all schools with a curriculum in which design is a

fundamental element of the academic program. This starts from the middle of the second

grade of the Artistic Liceo (high school), or the Art Institutes, Technical or ProfessionalInstitutes, up to the degrees of laurea (i.e. Bachelor's, Master's degree) in Architecture,

and Engineering, Civil, Structural, Environmental & Mechanical. Parabolas, hyperbolasand especially ellipses, have posed a problem any time one needs to represent a sphere,

cylinder, cone, circle, and all their respective intersections with planes on the two

dimensional surface of a sheet of paper. Tracing a conic curve in general consists of theinvention of a curve satisfactory to the eye through an array of a few distinct points,

perhaps even some tangents.

23) The most common problem in tracing ellipses is given by the perspective

representation of the circumference inscribed in the square; that is, the need to define the

inscribed curve in a general quadrilateral. In projection, the problem is equivalent todetermining the sections of an oblique cone having a circular generatrix. The center of

projection represents the vertex of the cone, the generatrix of which are the projecting

radii which extend from the center of projection to the circular contour of the figure one

wants to render. The plane of the square will section the radii, generating the conic

section.

24) The first elements of a circle (not shown) usually represented in perspective, are the

image G'of the center Gand a pair of axes which are orthogonal at first with the relativevertices, the local tangents of which form the image of a circumscribed quadrilateral.

Neglecting the different systems of possible solutions, the result of the first graphicaloperation is a quadrilateral, as shown in Fig. 9. The two pairs of opposite sides converge

in the two vanishing points Faand Fb, belonging to the vanishing linefof the plane of

. The diagonals meet in the point G', and the lines going through G' and vanishing

points Faand Fbare the image of two orthogonal axes of circle . These determine on thecorresponding sides, two pair of points which are the images of the orthogonal vertices of

. By extending the diagonals we obtain onf the relative vanishing points Fcand Fd.

25) By the law of the invariance of the cross

ratiounder projection and section [Note 9],cross ratio (Fa , Fb; Fc , Fd)is equal to (a3,

b3 ; c , d) as well as the other cross ratios

such as (Fa , A1 ; D1, C1 ), of the quartet of

intersections of the radial array of lines oncenter G', that is, those belonging to the

sides of the quadrilateral. (Fa, Fb; Fc, Fd)is

also equal to the cross ratio (f , b3 ; b1, b2)of the array of lines on center Fb,because it's

projective with the harmonic quartet FaA1D1C1and is therefore equal by projection

to the cross ratio (Fa , G' ; B1 , B2), and thequartet of points aligned to our completequadrilateral in the plane (the quadrilateral

formed by the three pairs of lines - the sides

and the diagonals).

26) The three vertices of the harmonic

collection G', FaandFb , common points ofthe three pairs of lines of the complete plane

quadrilateral, determine the diagonal

triangle,on the sides of which are the pairsof conjugate points that are projections of


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pairs of conjugate rays.

27) Each one of the two sides of thediagonal trianglewhich cut the conic do so

at the tangent points of the sides belonging

to the radial array of lines of the oppositevertex. Therefore,A1andA2of b3are tangentpoints to the ellipse of the lines a1and a2of

the radial array on Fa , and B1 and B2 are

tangent points of the lines b1 and b2 of the

radial array on Fb.

28) If we next consider the harmonic

quartets A1A2 G' Fband B2B1 G'Fa,we findthat a polarity relation exists between

vanishing point Fb and line a3, and another

polarity relation exists between vanishingpoint Faand the lineb3,both with respect tothe conic. Points B1 and B2 are the tangent

points of the lines b1 and b2, respectively,

converging in the pole Fb, and a3is thepolarline of the conic with respect to Fb.

Similarly, between vanishing point Fa and

the line b3, A1andA2are the tangent pointsof lines a1and a2, converging in pole Fa, and

line b3 is the polar line of the conic with

respect to Fa.

29) If, in the quartetA1A2 G'Fb,we substitute for G' the median pointA0of segmentA1A2keeping the cross ratio constant, the conjugate of A0 will be the ideal point of b3. See Fig.10. As a result, in order to preserve the cross ratio for projection and section, the same

vertices projected from Fa will generate the harmonic quartet of the pair of conjugatelines a1 a2 a0fb3. The line a0has a conjugate direction with respect tofb3and goes through

A0, so it's an axis of oblique symmetry and a diameter of the conic.

30) To obtain the center of the ellipse, we now determine another diameter. In the second

quartet B2B1 G'Fa we substitute at G' the point B0, which is the median of B1B2). Its

conjugate will be the ideal point of a3 see Fig. 11. For the projection of the verticesfrom Fbwe generate the harmonic quartet b1 b2 b0fa3 , and b0 is another axis of oblique

symmetry of the conic.

31) The intersection of a0 and b0 gives thecenter Oof the ellipse. Using the vertices of

the quadrilateral we can then distinguish

four other pairs of conjugate diameters toconfirm or replace the preceding ones. If

two sides of the quadrilateral are parallel, as

well as the two sides of the diagonal

triangle, as shown in Fig. 12, the side of thistriangle determined by its two ordinary

vertices will constitute an axis of oblique

symmetry of the ellipse, since it will dividein half the parallel segment of the triangle

between the non-parallel sides of the

quadrilateral (half of this segment is

highlighted in the Fig. 12).

32) If the two external vertices of the

diagonal triangle are ideal, the internal

vertice will coincide with the geometriccenter of the conic, and the two sides

relative to it will then be diameters. The


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quadrilateral will be a parallelogram and its sides will be parallel to the two diameters

that are the two median axes, as shown in Fig. 13. Obviously, this is the classic case ofrepresentation of a circle or ellipse in axonometry, or more generally in parallel

projection. In fact its frequently the case that the representation of circles (as ellipses) in

orthogonal projection will require this type of construction.

Note 1

35) The point conic. Consider two distinct radial arrays of lines (in the plane) through

points Uand U. See Fig. 14. Between these two points, define aprojectivityusing the

one-to-one correspondence of at least three corresponding points or lines. Lets takethree pairs of lines (a, a), (b, b), (c, c). The intersection of each pair of lines is called a

common point(shown as larger black dots on the dotted ellipse). Each two pair of linesdetermine a quadrilateral (or projectivequadrangle). Fig. 14 highlights the

quadrilaterals [(a, a), (b, b)] and [(b, b), (c,

c)]. The third quadrilateral [(a, a), (c, c)]

extends off the page to the right. Within each

quadrilateral, the intersections of each line and

its partners prime (e.g. (a, b), (a, b)) are

called the associate points (shown as smallblack circles). The associate points are always

aligned to a point O, called the center of

projectivity.

36) When the center of projectivity Ocoincides

with the line containing points (U, U), the lineis called united because it corresponds to itself

in the projectivity, which then becomes aperspectivity. When U and U coincide, the

projectivity becomes an involution, and thecorresponding elements are called conjugate.

37) The first theorem of Jacob Steiner (1796-1863) states that: if from two points Uand

Uwe project the other points, and if we retain two corresponding rays that project the


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same point, we will have established a projectivity between U and U, the center Oofwhich is the intersection of the tangents of the conic through Uand U.

38) Given five points, its possible to construct

the conic as a result of the above projectivity

between the two radial arrays of lines. See Fig.15. Considering the two points A and B as

centers, we project from the other three pointsC, D, andE,rays c, d and ethroughA, and c, d

and e through B. Once weve distinguished

the associated quadrilaterals, we determine the

position of center of the projectivity O thoughthe associate points, and the tangents a and b of

the two centers A and B. For every new ray

from A there corresponds a ray from B. The

geometric locus of the resulting common points

(as defined above) generate the conic werelooking for.

38b) Now lets take a closer look at the

mechanics of the projective correspondence

between lines (or rays) on polar points A andB- see Fig. 15b. A new ray x from A must

cooperate with any existing pair of rays though

one of the generating points C, D, andE -either (c, c), (d, d) or (e, e) in the followingmanner. Suppose we select the pair of rays (c, c). Rayx fromA, in any direction not

parallel to ray c from B, will intersect cat some finite point xc.This is the first newassociate point. Ray Oxc from the projective center O to point xc, will then intersect

ray cfromAat some point xc. This is the second associate point, and it defines ray x

from B. Rays x and x then define the new common point xx by their intersection.

Holding pointsA, B, C, D, E, O fixed, if either rayxorx is rotated around its respectivepole (A or B) while maintaining the projective relationship between xandx, then point

xxwill sweep though the path of the complete conic, thereby defining it. In fact, raysx

and xwill simultaneously cooperate with allpairs of rays through C, D and E, in themanner just described.

39) To determine the local tangent at any point of the conic, its sufficient to determine

the projectivity that has this point as one of the two centers of the radial arrays (or poles,


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such asA orB), and then to connect this point with a line to the center of projectivity O,

that weve established.

40) If we know one point (such asA) with a tangent (such as a) going through it, three

more points (such asB, E, D) will be sufficient to generate the conic. If we consider the

tangent point as one of the poles, then its possible to determine the center of projectivityby tracing a single quadrilateral projective to two pairs of corresponding rays (such as d,

d, e, e), tracing the diagonal through the associate point and finding its intersection

with the tangent (such as a). Then its possible to determine additional points of theconic. As a result, its enough to know these three points and the tangents through two of

them, in order to determine the conic.

Note 2

41) The line conic. By using the principle ofduality,we can move from consideration of the

point conics to the line (or envelope) conics. Inplace of radial arrays of lines through polar

points, we now consider two distinct lineararrays of points, u and u, between which a

projectivity can be defined as follows see Fig.

16. We define as correspondents the three pointpairs (A, A), (B, B) (C, C). Each point pair

defines a common line containing both points,

that isAA, BB, CC. Each pair of lines - (AA,BB), (AA, CC), (BB, CC) defines a

quadrilateral in conjunction with the lines u andu, the diagonals of which are called associate

lines. The associate lines of a quadrilateral

always intersect on a line s, called the axis of

projectivity.

42) In the special case where s coincides with

the intersection of u and u, this point is calledunited, because it corresponds to itself in the

projectivity, which then becomes a

perspectivity. If the two sets of points coincide,the projectivity becomes an involution.

43) The fourth theorem of Steiner states that: The lines common to two linear arrays of

points that are projective, but not perspective, form the envelope of a unique conic. This

includes the intersection of the axes of projectivity with the sets of points uand u, which

are also the local tangents.

44) Given five lines, its possible to construct a conic as a result of the projectivity

between two linear arrays of points. After selecting two lines u, u as projective arrays ofpoints, we can determine the other lines by determining the corresponding points A, B,

and C of u, and A, B and C of u. After determining just two out of three possible

projective quadrilaterals of corresponding pairs of lines, we can use the intersection of thediagonals (associate lines) to determine the axis s of the projectivity. Tangent points S

and S of the conic are determined by the intersections of swith u and u. For every newpoint of u there will be a corresponding point on u, and the new common lines so

defined will comprise the envelope of the conic relative to the projectivity.


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44b) Now lets take a closer look at the

mechanics of the projective correspondencebetween points on lines u and u - see Fig.16b. A new linexon umust cooperate with

any existing pair of points (A, A), (B, B) or

(C, C) generating the projectivity and thelines a, b, or cthey define - in the following

manner. Suppose we select the pair of

points (B, B). RaysXB fromX toB, in anydirection not parallel to polar line s, will

intersect sat some finite pointX. LineXB

is the first new associate line. Line BXfrom point B to line s at point X, is then

extended to intersect line u, generating point

X. Line BX is the second associate line,

and point X is now the intersection of

associate lines XB and BX. Holding linesa, b, c, u, u, and s fixed, if either pointXorX is translated along its parent line (u oru) while maintaining the coincidence of

pointX with polar line s, then linexwill be

tangent to the conic as it rotates around it,thereby completely defining its envelope.

In fact, linexwill simultaneously cooperate with allpairs of points (A, A), (B, B) and(C, C) defining lines a, b and c, in the manner just described.

45) If you know one point of the conic and the tangent through it, then three moretangents will be sufficient to trace the conic. Considering the tangent through a known

point (such as S or S), as one of the lines (linear point arrays), its possible to determine

the axis of the projectivity by tracing a single projective quadrilateral of two pairs of

corresponding points, then to trace the associate lines, then connect the intersections withthe known point. In this way, its sufficient to know three tangents and two points of

tangency of two of those lines.

Note 3

46)Pascals theorem. According to the theorem of Blaise Pascal (1623-1662): Given six

points belonging to a conic, if we build a hexagon using any sequence of [these points as]

vertices, the lines of the opposite sides (i.e. non-consecutive) meet in three colinerar

points. The common line is called the Pascal line.See Fig 24b. Based on six points,A,

B, C, D, Eand F, we can compose different sequences of corresponding hexagons, each

one with a different Pascal line. On six vertices of a conic the simple combinations of

possible sides, that is pairs of points defining a side, are given by the coefficients of this

binomial expression:

C6,2= (6! / 2!(6 2)!) = 15

These are the sides 1 (AB), 2 (AC), 3 (AD), 4 (AE), 5 (AF), 6 (BC), 7 (BD), 8 (BE), 9

(BF), 10 (CD), 11 (CE), 12 (CF), 13 (DE), 14 (DF), and 15 (EF)shown in Fig. 24c.

The same thing can be said for the combination of four vertices:

C6,4= 15

These are the pairs 1 (AB + CD), 2 (BC + DE), 3 (CD + EF), 4(DE + AF), 5(AB + CE),

6(BC + DF), 7(CD + EA), 8(DE + FB) , 9(AB + CF), 10 (BC + EF), 11 (CD + AF),

12 (AC + EF), 13(AB + DE), 14(AB + DF), and 15 (AB + EF)shown in Fig. 25. For

each pair of opposite sides there is a corresponding point through which passes a Pascal

line. For example in Fig. 24b, point P1 corresponds to pair 10 (BC +EF),point P2 to

pair 11 (CD + AF), and point P3to pair 13 (AB + DE).


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Through any four vertices, we can trace three different pairs of lines, such as the pairs 1

(AB + CD), 2 (BC +DA), and 3 (AC + DB) shown in Fig. 17. Each pair of lines has an

associated point of intersection (1, 2, 3).


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Through any six vertices on a conic we can define a hexagon. Keeping constant a couple

of opposite sides, we can trace four different hexagons with four different Pascal lines,using four different sequences, such as the sequences 1 (ABECDF), 2 (ABEDCF), 3

(BAECDF) and 4 (BAEDCF) shown in Fig. 26. See also Fig. 18, which shows the four

sequences superimposed, along with their associated Pascal lines (in a different

arrangement of verticesA-F).

Thus, for every six distinct points on a conic curve we can define (15 x 4) = 60 distinct

Pascal lines.

Note 4

47)Brianchons theorem. The dual theorem to Pascal's theorem is that of Charles-JulienBrianchon (1785-1864) according to which: Given six points on a conic you can

construct a hexagon composed of the tangents to the conic though those points, and

collectively the lines that connect opposite

vertices of this hexagon will intersect in a

common point; this point is called the

point of Brianchon. Analogously to the

theorem of Pascal, for every circumscribedhexagon there corresponds a point of

Brianchon.

Note 5

48) Determining a chord. Given fivepoints A, B, C, D and E, we want to

determine the length of a generic chord

going through one point (for example,

chord athrough pointA).

We can determine any sequence with the

unknown point Xthat precedes or followsthe known vertices, for example (XABCDE)

as shown in Fig. 19. The first point P1 isdetermined by the opposite sides AB and


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DE, the second, P2, by the opposite sides CDandXA. Even if for the latter side (XA) we

don't know the second point X, but only the chord a, the line of Pascal p is stilldetermined. The third point P3is determined by the side BCand the Pascal line, so we

can trace the side (E P3) and determine in this

way the unknown pointXof the conic.

Note 6

49)Determining a tangent. Given five pointsA, B, C, D, and E, we want to determine the

tangent through one point (for example,

tangent a through pointA).

We can establish any sequence in such a wayto consider a sixth point coincident with A,

such as (AABCDE), as shown inFig. 20. Thus,

as this sixth pointX(not shown) approachesA,the side AX approaches tangent a. The first

point P1 of the Pascal line is given by the

opposite sides AB and DE, the second,P2, byBC and EA. After tracing the Pascal line p

which connects (P1 P2), the third point P3 is

given by the intersection of CD with p. The

tangent ais the line connecting P3withA,thelimit of sideAXof the hexagon, when points A

andX merge.

Note 7

50)Determining a pair of conjugate diameters of a conic. Given five points we want to

determine a diameter and the direction of its

conjugate.

We can establish any sequence with the sixth

unknown pointXthat precedes or follows one of

the vertices, for example (ABCDEX ),and tracing

the direction of the chord AX parallel to theopposite side CD in the sequence, as shown in

Fig. 21. The first point P1 is given by the

intersection of the opposite sides AB and DEwhile the second point P2is ideal (at infinity)

because it's the intersection of the parallel lines

CD and AX. The Pascal line p is parallel tothose lines (CD andAX) passing through point

P2. The point P3is determined onpbyBCand

consequently X is determined by EP3 on thechord passing through A. The diameter will

bisect the two parallel chords AX and CD, and

has a direction conjugate to them. Thus weve

completely determined one axis (a), anddetermined the direction of its partner, parallel

to the Pascal linep.

Note 8

51) The method of transformed points. Given thexandyaxes and the circles rxand ry,

we can trace two new orthogonal axes m, n, at some general angle with respect to the xandyaxes, as shown in Fig. 22. We then project the intersection points of mand nwith

each circle, in a direction perpendicular to the respective x or yaxis, but with opposite

sense of direction. That is, the point of intersection with the inner circle will be projectedperpendicularly to the minor axis and towards the outside, while the intersection point

with the outer circle will be projected perpendicularly to the major axis and towards the


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inside. The intersection of the pair of

projecting lines will occur in the annulusbetween the two limiting circumferences.

The four resulting points representtransformed points of both inner and outer

circles onto the ellipse. That is to say, a oneto one mapping is now established between

the points of each circle onto the ellipse.

Furthermore, this method can be used totransform axes, as well. For example, the

vertices of the conjugate diameters s and t

regulate the transformation to theirorthogonal counterparts s and t, as shown.

Finally, the four lines v, tangent to the

ellipse at s and t, are transformed images

of perpendicular tangents vrx and vry (not

shown) to the inner and outer circlesrespectively, at their intersections with s andt. In fact, tangents v form a parallelogramimage of unique squares formed by tangentsvrx and vry, circumscribing circles rx and ry

respectively.

Note 9

52) The cross ratio. On a point conic f , given a reference point Fc, and a positivedirection, for example, to the right, the ratio between the distances of points FaandFb

fromFc, called the ratio of division (or simple ratio), and denoted(FaFb, Fc), is definedas:

(FaFb, Fc) = (FaFc) / (FbFc).

Thus, we may determine the ratio of division between the lines of the radial array havingG' as the center, such as a3, b3, c. In this case, it can be shown that the followingrelationship exists between the sines of the angles formed by the first two rays with

respect to the third:

(a3b3, c) = sin (a3 c) / sin (b3 c), wherea3 c and b3 cdesignate the angles formed between

line pairs(a3,c) and(b3,c), respectively.

We now take the two points Fc and Fd as a reference pair, and consider the ratios ofdivision that connect them to the relative pairFaand Fb, that is, (FaFb, Fc) and (FaFb,Fd). The ratio between these ratios of division is calledthe cross ratio (double ratio, or

bi-ratio), is designated (Fa, Fb; Fc, Fd), and definedas:

(Fa, Fb; Fc, Fd) = (FaFb, Fc) /( FaFb, Fd) = {( FaFc) / (FbFc)} {( FbFd) / (FaFd)}

Similarly, for the rays of the radial array on center G' we have the cross ratio (a3, b3; c,

d). Setting (Fa, Fb; Fc, Fd) =x,it can be shown that:

i) Interchanging the position of any two of the four points simultaneously with the

other two points won't change the cross ratio, and exchanging the points of onlyone pair (either the relative or reference pair) will change the cross ratio into itsreciprocal (1/x). If we exchange only the inner two or outer two points, the cross

ratio will change into it's arithmetic compliment (1-x);

ii) If we section four rays of a radial array, the points of intersection (with thesection) will have the same cross ratio. The cross ratio will not change in

projection or section, that is, its invariant;

iii) If the cross ratio is equal to (-1) the quartet is called harmonic, like the onewereexamining: (Fa, Fb; Fc, Fd);


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iv) In a harmonic quartet if one element is ideal, its harmonic conjugate is the

median of the segment defined by the other pair in the quartet;

v) In a harmonic quartet every element of a pair is called conjugateto the other

element in the pair, because of it's correspondence in the involution between

superimposed forms. Therefore, each pair in a harmonic quartet is formed by twoconjugate elements.

If, in the harmonic quartet, we trace two circles having as diameters, respectively, therelative point pair (FaFb) and the reference point pair (FcFd),these circles will intersect

at a point O*, through which will pass the circle with diameter formed by the two centersof the first two circles. The angles at the vertices (FaO* Fb) and (FcO* Fd) will be right

angles, with the pair of angles rotated 45 with respect to each other, and for which each

angle will be bisected by a side of the other angle, as shown in Fig. 23.

5 Point Ellipse

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