A generalization of the polar theory to egg line and egg surface

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E. Gergely
Institutul de Calcul

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E. Gergely, Eine Verallgemeinerung der polaren Theorie auf Eilinie und Eifläche. (German) Mathematica (Cluj) 1 (24) 1959 221–237.

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Mathematica Cluj

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Published by the Romanian Academy  Publishing House

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1222-9016

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2601-744X

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A GENERALIZATION OF POLAR THEORY TO ICE LINE AND ICE SURFACE

E. GERGEIN

Cluj

The theory of plane convex figures, and likewise the theory of convex surfaces and solids, is an important and large field of mathematical research. This work deals with the generalization of polar theory to an important class of convex figures and convex surfaces, namely egg-shaped figures and egg-shaped surfaces
We can realize this generalization in several ways, using various characteristic features of polar kinship, naturally such properties which allow a generalization to egg lines and egg surfaces.
In the following, we will only consider two possibilities of this generalization.
In the first part, we define the polar curve in the case of the oval, and the polar surface in the case of the oval surface, using harmonic pairs of points. In the second part, only in the case of the oval, we will allow the points of the diagram to correspond to the so-called "conjugate curves." These curves are the geometric loci of the points of intersection of the tangents at those points on an oval where the lines through the pole intersect the polar curve. These curves are also generalizations of the polar curves in the case of conic sections. This generalization is not possible in the case of the oval surface.

§ 1 Polar curves and conjugate curves of the oval lines

In the following, we understand an oval curve to be a convex, closed curve that has a definite tangent at every point. We can note that the convexity of the curve implies a monotonic change in the direction of the tangent and also the property that the curve has no corners.
The oval curve is a Jordan curve and, as such, divides the points on its plane into inner and outer points. For the outer points, we can generalize the concept in various ways.
I. Two definite tangents pass through an outer point to the oval curve; the line through the points of tangency of these tangents we call the polar line of that point.
II. Let us consider such a line a through the outer point P 1 P 1 P_(1)P1which intersects the oval. The polar curve of the point P P PPis the geometric locus of the point which P P PPfrom the points of intersection of the line a with the curve of the egg, it is harmonically separated. This determination also makes sense in the case of an interior point with respect to the curve of the egg, and in this way all interior points also have a polar curve.
The term I only has meaning for the outer points of the oval line,
III. We determine the conjugate curve of a point. P P PP(inner or outer) as the geometric locus of the points of intersection of the tangents at the common points of the oval line and the line through the point P P PPThese curves are also generalizations of the polar concept for conic sections.
In the following, we will deal with cases II and III.

1. Polar curves.

a) Polar curves of an external point.
Fig. 1
It is necessary to introduce some new terms.
The triangle with sides a a aa, and b b bbTangents of the oval line through the outer point K K KKand the tendon through the points of contact A A AAand B B BBof this tangential line, we call the tangent triangle of the point K K KK(Figure 1).
The points A A AAand B B BBdivide the oval into two arcs; one is inside the tangent triangle, the other outside. We call this inner arc, and the outer arc the arc for the point K K KK.
Let us take all such straight lines through the point K K KK, which intersect the oval. These straight lines also intersect the chord A B A B ABABThe intersection point of such a straight line with the inner arc is... C C CCand the harmonically corresponding point to the pair of points K D K D KDKDlet E E EE. The set of points E E EEwe call the harmonic curve of the inner arc
The harmonic curve together with the inner arc gives an oval shape in the projection plane; the polar curve of the point. K K KKWith regard to the filine, the tendon A B A B ABAB.
Fig. 2
Proof: We show that a straight line g g ggintersects this harmonic curve at most at two points. Indeed: one point of intersection through our construction corresponds to a single point on the inner arc and the straight line g g ggcorresponds to a straight line g ′ g ′ g^(')g′; this g ′ g ′ g^(')g′can intersect the convex inner arc at most two points and therefore it is also true for the harmonic curve
To study the polar curve, it is necessary to study the relative position of two oval lines, especially the case of a circle and an oval line.
An oval and a circle have an even number of points of intersection, but they can also share some arcs. We will exclude this last case. Let us observe all circles in the plane of the oval. The maximum number of points of intersection of a circle with the oval is called the order of the oval with respect to the circle.
For an even number 2 n 2 n 2n2nThere always exists an egg line with order ≧ 2 n ≧ 2 n >= 2n≧2nRegarding the circle, let's consider a circle. 2 n 2 n 2n2nPoints. It can be seen that it is possible to construct an oval that intersects the circle at these points, and whose arcs between the points of intersection lie alternately inside and outside the circle. The order of this oval, with respect to the circle, is greater than or equal to 2 n 2 n 2n2n.
We can use the order with respect to the circle for a classification of the oval lines, corresponding to the concept of the order of a general curve with respect to the straight lines of the plane, where the order is the maximum number of intersection points of the curve and the straight line.
A fine line and a straight line have at most two points of intersection, and thus the order—in this sense—of an oval is 2, and further classification in this way is not possible. The order of an ellipse with respect to a circle is 4; this number is the minimum order of an oval with respect to a circle.
Similarly, for an oval line, we can construct other oval lines which intersect the given oval line at a given even number of points.
According to this, we can also study the shape of the polar curve using the harmonic curve.
There are three possible cases:
a) the harmonic curve lies entirely within the curve of the egg
; b) the harmonic curve lies entirely outside the curve of the egg;
c) the harmonic curve intersects the curve of the egg, or the two share a common arc.
In case a), the polar curve lies entirely inside the area defined by the chord. A B A B ABABand the outer arc of the egg-shaped area limited and does not cut the tendon A B A B ABAB(except for the points) A A AAand B B BB). (Figure 2).
In case b), the Po-
Fig. 3
Lar curve inside (except for the points) A A AAand B B BB) of the tangent triangle.
In case c), we must distinguish 3 possibilities with regard to the relative position of the oval line and the arc. O 1 O 1 O_(1)O1(Figure 3):
c 1 c 1 c_(1)c1) the points of intersection of the oval line O O OOwith the bow O 1 O 1 O_(1)O1;
c 2 c 2 c_(2)c2) the points of contact of the oval line O O OOand the arc O 1 O 1 O_(1)O1;
c 2 c 2 c_(2)c2) the common arc of the egg line and the arc O 1 O 1 O_(1)O1.
The points of intersection of curves 0 and O 1 O 1 O_(1)O1correspond to certain points on the polar curves; these points are the points of intersection of the chord A B A B ABABand the polar curve.
The polar curve touches at points which correspond to the points of the case c 2 c 2 c_(2)c2correspond to the tendon A B A B ABAB(possibly at inflection points). In the case c 3 c 3 c_(3)c3, a part of the chord A B A B ABABis shared with the polar curve. In this case, the polar curve, together with the inner or outer arc, does not form an oval shape
This is generally true also in cases a) and b).
Let us denote by α α alphaα(Figure 4) the angle between t 1 t 1 t_(1)t1and c c ccand with x x xxthe oriented segment of the line c c ccbetween the outer arc and the arc O 1 O 1 O_(1)O1(the positive direction from the point K K KKafter the outer arc). The function
Pig. 4
tion x ( α ) x ( α ) x(alpha)x(α)is continuous and partially monotonic, or constant. The monotonic parts of the function x ( α ) x ( α ) x(alpha)x(α)The corresponding arcs are concave or convex with respect to the outer arc.
A polar curve has a tangent at every point. We can easily construct this tangent (Fig. 5).
Let's take the outermost point. K K KKand its polar curve; P P PPa point on the polar curve. K C D K C D KCDKCDis a line K P K P KPKPdifferent straight lines; Q Q QQis the point on the polar curve on the line KCD. The lines A C , P Q A C , P Q AC,PQAC,PQand B d B d VolBdmeet at one point T T TT; the routes A C A C ACACand B D B D BDBDare tendons of the filine and P Q P Q PQPQa chord of the polar curve. If the straight line K C D K C D KCDKCDthe straight line K A B K A B KABKABAs the lines approach, they approach each other. A C T A C T ACTACTand B D T B D T BDTBDTthe tangents at the points A A AAand B B BBand so the point approaches T T TTthe intersection point of the tangents (in the special case these tangents are parallel and then point T T TTto infinity). The straight line P Q P Q PQPQtherefore approaches a well-defined straight line and 50 50 ^(50)50The polar curve has a well-defined tangent at all its points.
Pig. 5
Fig. 6
The construction of the tangent is therefore very easy; we construct the tangents at the points of intersection of the lines K P K P KPKPand the oval curve; the tangent to the polar curve is the one passing through the point P P PPand the straight line determined by the intersection of the tangents.
It is now clear that the change in the direction of the tangent to the polar curve is continuous
THEOREM. A single polar curve passes through all the interior points of the oval in every direction.
In other words, for all line elements belonging to an interior point, we can define a single exterior point such that for the polar curve of this point, the given line element is a tangent element.
Proof: (Figure 6). Let P P PPan interior point of the oval and a a straight line through the point P P PP. We denote the tangents to the oval line parallel to a by . b b bband c c ccand their points of contact with B B BBand C C CC; the points of intersection of the line a a aawith the oval with M M MMand N N NNand finally the tangents at these points with m m mmand n n nn. The point S S SSruns along the straight line from infinity to point M. Let us draw the chords of the oval through the points of tangency of the tangents S S SSThese tendons simply cover the inside of the oval line between the tendon and the point. M M MMIf the point S S SSon the straight a a aain the opposite sense from infinity to N N NNIf the string runs, then the corresponding chord simply covers the interior from the chord to point N. So, if the point S S SSthe straight a a aa- with the exception of the route M N M N MNMN- passes through, the corresponding tendons simply cover the entire inside of the oval line, and so it goes for a single point S S SSthe corresponding tendon through the point S S SS. We therefore have the existence of a through the point P P PPgoing
and the straight a a aatouching polar curve and because a a aaIf any straight line passing through the point is the endpoint, we have proven the first part of our theorem. We denote the endpoints of the chord thus constructed by D , E D , E D,ED,E. The point K K KK, whose polar curve through P P PPgoes and the straight a a aaat the point P P PPtouches, forms with P P PPregarding the points D , E D , E D,ED,Ea harmonic pair of points; consequently, the point K K KKand thus the polar curve in question is completely determined, and therefore our theorem is completely proven.
b) The polar curve of an interior point.
It is clear that the polar curve of an interior point B B BBentirely in
Fig. 7
outside the oval. Through the point B B BBpass those chords whose midpoint is precisely the point B B BBThe straight lines determined by such curves are the asymptotes of the polar curve of the point. B B BBAnd so the polar curve of an intrinsic point has as many points at infinity as there are such chords passing through it whose midpoint is the given point.
The polar curve of an interior point has at least one point at infinity; therefore, such polar curves have at least one branch extending to infinity (Figure 7). A curve passing through the point B B BBThe percutaneous tendon has two parts. A B A B ABABand B C B C BCBC. Let it be A B >> B C A B >> B C AB>>BCAB>>BC. By rotating around B B BBThe lengths of the segments change constantly. This is due to a rotation at an angle. π π piπthe line segment B C B C BCBCtransitions into A B A B ABABand thus there exists at least one position of the chord in which B B BBthe midpoint. Therefore, on the line defined by this chord, the point is... B B BBcorresponding conjugate point. A point on the polar curve at infinity.
Infinitely many polar curves pass through all exterior points. (Figure 8). It is clear that the polar curves of the points which lie on the polar curve of the point K K KKlie, through K K KKgo. The construction of the tangent to a polar curve at an external point is as follows; we assume that the polar curve to the point B B BBheard. The points of intersection of the lines. K P K P KPKPwe denote with M M MMand N N NNand the intersection of the tangents of the oval line at the points M M MMand N N NNwith T T TT. The tangent we are looking for is the straight line determined by K K KKand T T TT. We can prove this construction in a similar way to how we proved a polar curve through an interior point
Eig, S
Fig. 9
Fig. 10 a
Lemma. (Fig. 9) Let K be the harmonic point of the interior point B with respect to the points of intersection AC of the line KB with the oval, and M M MMThe intersection point of the tangents at A and C. The line MK
does not intersect the oval. Truly, the oval and the point K K KKcannot lie in the vertical angle formed by the lines MA and MC, and therefore MK cannot intersect the oval line.
The straight line M K M K MKMKis the tangent at points K K KKof the polar curve of the point B B BBand in this way we proved that the tangents of the polar curves are given by K K KKThey cannot cut the egg line. Consequently, they pass through it. K K KKThe tangents drawn through the outer point do not determine polar curve tangents at the angle in which the oval line lies, and thus we cannot draw polar curve tangents in all directions through an outer point, as in the case of an inner point.
In the following, we will prove that through an external point K K KKin all directions whose defining line does not intersect the oval line, a single line in question at the point K K KKtouching polar curve.
We introduce the concept of the conjugate curve of a point with respect to the oval. The conjugate curve is the geometric locus of the points of intersection of the tangents drawn at the endpoints of the chord passing through the point in question. We will consider the conjugate curve of an exterior point. K K KKstudy (Figure 10a). We are to prove that a K K KKA straight line that does not intersect the egg curve. e e eethe conjugate curve of the point K K KKat a single point C C CCcuts. In this
Fig. 10b
case - which is formed by the points of contact of the c c ccgoing tangents going straight α α alphaα. through K K KKLet D D DDbe the harmonically conjugate point of K K KKon the line a with respect to the oval. The polar curve of the point D D DDgoes through K K KKand its tangent is precisely the straight line ε ε εε.
From a point on the line e e eewe construct the two tangents to the oval. (Figure 10b). The line passing through the points of tangency of these tangents intersects the line e e eeat that point. te Q ′ Q ′ Q^(')Q′. It is clear that one point Q Q QQa single point Q ′ Q ′ Q^(')Q′corresponds. We refer to this point Q Q QQas the conjugate point of the point Q Q QQon the straight c c cc.
One remark. The one between the dots ? and Q ′ Q ′ Q^(')Q′Existing correspondence is involutory in the case of an ellipse, but is generally not in the case of an ovary
We will call the conjugate point of the point after this. Q ′ Q ′ Q^(')Q′with Q ′ ′ Q ′ ′ Q^('')Q′′, the conjugates of the point Q ′ ′ Q ′ ′ Q^('')Q′′with Q ′ ′ ′ Q ′ ′ ′ Q^(''')Q′′′and so on. It is an open question whether this correspondence is involutory only for the ellipse for all points of all lines or not. If for the points Q , Q ′ , Q ′ ′ , … , Q ( n ) , Q ( n ) = Q Q , Q ′ , Q ′ ′ , … , Q ( n ) , Q ( n ) = Q Q,Q^('),Q^(''),dots,Q^((n)),Q^((n))=QQ,Q′,Q′′,…,Q(n),Q(n)=Q, then we call the sequence of points a chain of degree n. Many closure problems also arise in connection with this concept.
If Q Q QQthe straight e e eedescribes, then different points can be... and Q 2 Q 2 Q_(2)Q2not the same point Q ′ Q ′ Q^(')Q′correspond. Truly, let us observe the tendons which are formed by the points of contact of the points Q 1 Q 1 Q_(1)Q1and Q 2 Q 2 Q_(2)Q2drawn tangents are determined. If the points Q 1 Q 1 Q_(1)Q1and Q 2 Q 2 Q_(2)Q2a single point Q Q QQIf that were the case, then these two tendons would intersect at an outer point of the oval line, but this is not possible due to the continuous rotation of these tendons.
In this way, the correspondence of the points is Q Q QQand Q ′ Q ′ Q^(')Q′unambiguous. We still need to prove that the point Q ′ Q ′ Q^(')Q′the entire straight e e eedescribes when the point Q Q QQIt does.
Let the points A A AAand B B BBbe the points of contact of the with e e eeparallel tangents. The straight line A B A B ABABintersects the line c c ccat the point K K KK; we call this point the midpoint of the line with respect to the line. Let the point L L LLthe conjugate of the point K K KK. The direction from the point K K KKto L L LLLet's take that as the negative direction. The point Q Q QQDescribe the straight line e e eefrom the point + ∞ + ∞ +∞+∞to the point − ∞ − ∞ -oo−∞. To the infinite
Fig. 11
Fig. 12
Point on the line corresponds to the point K K KK. If O O OOin the interval [ + ∞ [ + ∞ [+oo[+∞, K ] K ] K]K]then it is clear that Q ′ Q ′ Q^(')Q′in the interval [ K , − ∞ ] [ K , − ∞ ] [K,-oo][K,−∞]lies.
Let us construct the conjugate curve of the infinity point of the line e e ee(Figure 11). This curve intersects the straight line. e e eein a single point K 1 K 1 K_(1)K1and the conjugate point of the K 1 K 1 K_(1)K1is the infinite point of the line e e eeWe refer to this point K 1 K 1 K_(1)K1the quasi-center of the line with respect to the oval (in the case of the ellipse, the points are K K KKand K 1 K 1 K_(1)K1identical, the point L L LLis the middle point of the line e e ee.
The location of the point Q ′ Q ′ Q^(')Q′, as a result of its construction, is a continuous function of the position of the point. Q Q QQ(except for the point K 1 K 1 K_(1)K1).
If Q Q QQthe interval [ + ∞ , K ] [ + ∞ , K ] [+oo,K][+∞,K]describes, then the point Q ′ Q ′ Q^(')Q′describes the segment of K K KKuntil L L LL; and if Q Q QQmoves within the interval K K 1 K K 1 KK_(1)KK1then describes Q Q QQthe interval [ L , − ∞ ] [ L , − ∞ ] [L,-oo][L,−∞], and finally, when Q Q QQthe interval [ K 1 , − ∞ ] [ K 1 , − ∞ ] [K_(1),-oo][K1,−∞]describes, then goes Q ′ Q ′ Q^(')Q′from + ∞ + ∞ +∞+∞to the point K K KK. Our result is that Q ′ Q ′ Q^(')Q′the entire straight e e eedescribes and thus we have proven the theorem concerning the directions of the polar curves passing through an external point
Some special ovarian lines
Interesting special ovarian lines are those that contain a so-called partial center O O OOhave, i.e., where a point exists in their plane for which some arcs of the oval are symmetrical, but the complementary arcs are not, (Figure 12) the arcs a , a ′ a , a ′ a,a^(')a,a′, respectively b b bb, b ′ b ′ b^(')b′are symmetrical, but c , c ′ c , c ′ c,c^(')c,c′, and d , d ′ d , d ′ d,d^(')d,d′, not). In this case, the polar curve of point O has no finite points in the angles containing the symmetrical arcs; in these angle sectors, the polar curve is an infinite straight line.
It is clear that we can construct such oval curves for which the polar curves of two
different points have points of intersection in the previously given number; there are even oval curves for which some polar curves have identical arcs. Consequently, there is no general theorem for the number of polar curves through two points. Indeed, if A A AAand B B BBtwo interior points and a a aaand b b bbif their polar curves are, then all their points of intersection pass through corresponding polar curves A A AAand B B BBWe saw, however, that the set of intersection points of two polar curves can consist of isolated points—in any number—and can even form continuous arcs. In this way, such pairs of points can exist in the plane of an oval curve through which a given number of polar curves pass.
Fig. 13
even their quantity can have the cardinality of the continuum.

2. The conjugate curves

We have seen that the conjugate curve of an exterior point passes through all points. K K KKintersects a straight line at a single point. In the following, we prove some simple but important properties of conjugate curves
The conjugate curve of an exterior point K K KKhas a single branch and this branch has a point at infinity. (Figure 13).
It is only necessary to state that through the point K K KKOnly such a straight line intersecting the oval is valid for which the tangents drawn at its points of intersection with the oval are parallel. We call a chord for which the tangents drawn at its endpoints are parallel a quasi-diameter of the oval.
Two quasidiameters intersect within the linea ova. Indeed, a quasidiameter a divides the linea ova into two arcs: o 1 o 1 o_(1)o1and o 2 o 2 o_(2)o2. It is clear that if b b bbis another quasidiameter, then one of its endpoints lies on the arc o 1 o 1 o_(1)o1, the other one on o 2 o 2 o_(2)o2, and so the quasidiameter goes b b bbin the strip which is determined by the tangents drawn at the endpoints of a, from one half-plane determined by a to the other half-plane, that is, it intersects the quasidiameter a a aainside the oval. In this way, the lines containing the quasidiameter simply cover the outside of the oval, and thus the stated property is proven.
The shape and position of the conjugate curve of an external point K K KKis indicated in Figure 14. Starting from the points of contact A , B A , B A,BA,Bthe tangents drawn from the point K K KKthe conjugate curve lies entirely outside the oval, it has a single asymptote and the direction of this asymptote is identical to the direction of the tangents drawn through the endpoints of the K K KKgoing quasidiameters.
The conjugate curve of an interior point B has as many branches extending to infinity as there are quasidiameters passing through the point. B B BBgo.
This property is evident.
Fig. 14
For a centrally symmetric oval (with center O), the conjugate curve of the point is O O OOthe infinite straight line.
The conjugate curves of two interior points intersect at a single point
Be the point M M MMthe intersection point of the conjugate curve G P G P G_(P)GPof the point P P PPand the conjugate curve G Q G Q G_(Q)GQof the point Q Q QQ. Then, according to the definition of the conjugate curve, the curve defined by the points of tangency of the curve defined by M M MMThe chord through the points is determined by the tangents. P P PPand Q Q QQgo and so the point M M MMis uniquely determined ( M M MMis possibly at infinity).
The more general theorem also applies
: The conjugate curves of two points intersect if and only if the line defined by the two points intersects the oval. Two conjugate curves can only have a single point of intersection.
One remark. These proven properties are analogous to the polar properties of the ellipse, observing that in the case of the ellipse, the conjugate curve - by its definition - is only the outer part of the polar line.

2. The Polar Theory of Egg Surfaces

1) The polar surface of an external point.

Let K K KKbe an outer point of the egg surface O O OO(Figure 15). We construct on a through K K KKgoing straight lines a the with K K KKRegarding the points of intersection of the line a with the egg surface, harmonic point C C CC: the geometric locus of the points C C CC, is the polar surface of the point K K KKA plane passing through the line a intersects the egg surface in an egg-shaped line and the set of polar curves of the point. K K KKThe polar surface is defined by this oval line.
Fig. 15
theorem: The polar surface of an external point has a well-defined tangent plane at all points
Let C C CCany point on the polar surface of the point K K KKWe denote the points of intersection of the line KC and the oval by A , B A , B A,BA,B. A plane passing through the line K C K C KCKCintersects the egg surface in an egg-shaped line. The construction of the tangent to C C CCthe polar curves of the point K K KKRegarding this oval line, we know that all tangents constructed in this way lie in a plane. The planes of tangency p 1 p 1 p_(1)p1and p 2 p 2 p_(2)p2of the egg surface at the points A A AAand B B BBintersect in a straight line b b bb(possibly at infinity). The tangents to the oval at the points A A AAand B B BBlie in the planes p 1 , p 2 p 1 , p 2 p_(1),p_(2)p1,p2and thus their point of intersection lies on line b. In this way, the tangents of the polar curves pass through the point of intersection of the oval plane with the line. b b bb, consequently all these tangents lie in the area of b b bband the point C C CCdeterministic plane; this is the tangent plane of the polar surface at the point C C CC.
Note that with the continuous change of the point C C CCon the polar surface, the points also change A , B A , B A,BA,Bconstantly changing and consequently also the levels p 1 , p 2 p 1 , p 2 p_(1),p_(2)p1,p2and so does the straight line b b bb; from this it follows that the change in the direction of the tangent plane of the polar surface is also continuous.
THEOREM: At an interior point, all surface elements are the tangent elements of the polar surfaces, and each surface element corresponds to a single polar surface.
In connection with this problem, we define the following: P P PP(Inner or outer) adjoint congruence of the line with respect to the egg surface. Through the point P P PPA set of lines intersecting the egg surface passes through this point; the set of lines of intersection of the tangent planes passing through the points of intersection of the egg surface with the line in question forms the congruence of lines adjoint to the point. Between the line passing through the point P P PPThere is a one-to-one relationship between the passing line and the line of congruence.
Let us study the view through an external point. K K KKlines of congruence. (Fig. 16) Let P P PPbe an interior point. The lines pointing to the point K K KKrelative tangent curve (the geometric locus of the points of tangency of the line originating from the point K K KK(tangent drawn to the egg surface), we call E E EE. The points of intersection of the egg surface and the points of the curve E E EEand of the point P P PPCertain tendons also describe a different curve on the egg surface. E 1 E 1 E_(1)E1and we call this the curve projected with respect to the point P P PPof the E E EECurve. The projected curve is the complement.
part of the curve E E EEand from the top P P PPcertain cones. Be C C CCan intersection point of the curves E E EEand E 1 E 1 E_(1)E1(if such intersection points exist). C C CCis a point on the curve E E EEand so the line intersects C P C P CPCPthe egg surface at one point C ′ C ′ C^(')C′of the curve E 1 E 1 E_(1)E1. The point C C CCalso lies on E 1 E 1 E_(1)E1And that's the point. C ′ C ′ C^(')C′also an intersection point of the curves E E EEand E 1 E 1 E_(1)E1In this way, the points of intersection of the curves can be determined. E E EEand E 1 E 1 E_(1)E1to be arranged into such pairs of points that the lines determined by a pair pass through the point P P PPgo. The tangent planes determined by such a pair of points intersect in a - through K K KKgoing - straight lines leading to P P PPadjoint line congruence.

In this way, so many
lines of congruence pass through the point K K KK, like half the number of intersection points of the curves E E EEand E 1 E 1 E_(1)E1.
For an outer point K K KKand for its tangent curve E E EEsuch inner points exist P P PPthe egg surface for which the projected curve is the curve E E EEcuts and those for which these the E E EEdoes not intersect. The sets of these points - we call them D D DDand D ′ D ′ D^(')D′- form regions inside the egg surface due to the continuity properties of the egg surface. The boundary of these regions is the geometric locus of those points for which the curve E E EEand their projected curves touch at least at one point, because they diverge from the curve at that point E E EEcertain part of the egg surface lie. For the points of the area D D DDand for its boundary points, the adjoint congruence is given by the point K K KKstraight lines, but for the points of the area D ′ D ′ D^(')D′The conjugate congruence does not have such a straight line.
In this way, an outer point causes the inner points of the oval line to be divided into areas.
We therefore possess a picture of the lines of the adjoint congruence.
One remark: With an analogous construction, we can for an arbitrary curve E E EEon the oval line - not for the tangent curves of the outer points - the projected curve E 1 E 1 E_(1)E1determine with respect to the interior points. The set of such interior points for which E E EEand E 1 0 , 2 , 4 , … , 2 n … E 1 0 , 2 , 4 , … , 2 n … E_(1)0,2,4,dots,2n dotsE10,2,4,…,2n…We call points of intersection D 0 , D 2 , D 4 D 0 , D 2 , D 4 D_(0),D_(2),D_(4)D0,D2,D4, ⋯ , D 2 n , … ⋯ , D 2 n , … cdots,D_(2n),dots⋯,D2n,…In this way, all curves result in a division of the inner points of the egg surface into areas, and through an analogous construction we obtain a division of the outer points
through an inner point B B BBmoving tendons h h hhcorresponds to a quantity k k kkof the line of congruence and the set s s ssof the planes is of the
k k kkand the point B B BBDefinitely. It's an open question whether s s ssall through the point B B BBincludes the passing levels, or in other words, whether in each passing point B B BBAre one or more lines of congruence located in the plane? (Figure 17).
Fig. 17
Let S S SSbe a plane passing through the point B B BB. Determine the planes of tangency passing through a straight line that does not intersect the surface of the egg, and the points of tangency of these planes, as well as the chord containing these points which defines the plane S S SSin one point T T TTthe intersection line of the egg surface with S S SSintersects. The set of points T T TTDue to the continuity properties of the egg surface, it is a simply connected, closed area whose boundary line is the egg line, i.e., the interior of the egg line.
So we have proven that in the plain S S SSat least one straight line c c ccof the line congruence adjoint to the point B B BBexists, i.e., at least one line with the following property: that which is defined by the points of tangency of the line e e eegoing planes of contact, a specific chord containing a straight line passes through the point P P PP. On the line containing this chord, we determine the harmonic point with respect to the surface of the egg, corresponding to the point B B BB. The tangent plane to the polar surface of this point is precisely the given plane S S SS.
In the following, we prove that at an interior point, each surface element is a tangent element only for a single polar surface. (Figure 18). The plane s s ssThe given surface element bisects the egg surface in an egg line. O O OO. Let's take all the outer parallel curves lying in the plane s s ssliegenden Parallelkurven O ( d ) O ( d ) O(d)O(d)of the egg line O O OO, where the d d ddThe distance between the two curves is... The plane s s ssdivides the egg surface into parts O 1 O 1 O_(1)O1and O 2 O 2 O_(2)O2Through the tangents of the curve O ( d ) O ( d ) O(d)O(d)We determine the tangent planes to the egg surface. It is clear that the point of tangency of one of these planes is in O 1 O 1 O_(1)O1, the other in O 2 O 2 O_(2)O2lies. The geometric locus of the points of contact thus constructed for a given O ( d ) O ( d ) O(d)O(d)consists of two closed continuous curves,
g 1 ( d ) g 1 ( d ) g_(1)(d)g1(d)transitions into O 1 O 1 O_(1)O1and g 2 ( d ) g 2 ( d ) g_(2)(d)g2(d)transitions into O 2 O 2 O_(2)O2. If d d ddfrom O O OOuntil + ∞ + ∞ +∞+∞grows, cover g 1 ( d ) g 1 ( d ) g_(1)(d)g1(d)and g 2 ( d ) , O 1 g 2 ( d ) , O 1 g_(2)(d),O_(1)g2(d),O1, respectively O 2 O 2 O_(2)O2of the ovary line O O OOup to the point C C CCand D D DD, which are the points of contact of the tangent planes parallel to s
Fig. 18
are. The curves g 1 ( d ) g 1 ( d ) g_(1)(d)g1(d)and g 2 ( d ) g 2 ( d ) g_(2)(d)g2(d)simply cover the surfaces O 1 O 1 O_(1)O1respective O 2 O 2 O_(2)O2. Truly, two curves g i ( d 1 ) g i ( d 1 ) g_(i)(d_(1))gi(d1)and g i ( d 2 ) ( i = 1 , 2 ) g i ( d 2 ) ( i = 1 , 2 ) g_(i)(d_(2))(i=1,2)gi(d2)(i=1,2)They cannot intersect or touch each other because at any point of intersection or touch – if such a thing existed – the egg surface has a well-defined tangent plane, and this plane s s ssintersects in a similarly well-defined straight line. The construction method of the curve g 1 ( d ) g 1 ( d ) g_(1)(d)g1(d)and g 2 ( d ) g 2 ( d ) g_(2)(d)g2(d)establishes a one-to-one relationship between its points. The tendons determined by corresponding points pass through internal points of the oval line O O OOdue to the convexity of the egg surface. For a certain d d ddthe points determined in this way describe a curve g 3 ( d ) g 3 ( d ) g_(3)(d)g3(d). If d d ddfrom O O OOuntil ∞ ∞ oo∞grows, the curves cover g 3 ( d ) g 3 ( d ) g_(3)(d)g3(d)simply the inside of the egg line O O OOand converge after the intersection point E E EEof the line C D C D CDCDwith the plane. A single curve passes through all interior points of the oval g 3 ( d ) g 3 ( d ) g_(3)(d)g3(d)If two such curves pass through a point g 3 ( d 1 ) g 3 ( d 1 ) g_(3)(d_(1))g3(d1)nnd g 3 ( d 2 ) g 3 ( d 2 ) g_(3)(d_(2))g3(d2)would go, then the corresponding g 1 ( d 1 ) , g 2 ( d 1 ) g 1 ( d 1 ) , g 2 ( d 1 ) g_(1)(d_(1)),g_(2)(d_(1))g1(d1),g2(d1)and g 1 ( d 2 ) , g 2 ( d 2 ) g 1 ( d 2 ) , g 2 ( d 2 ) g_(1)(d_(2)),g_(2)(d_(2))g1(d2),g2(d2)curves would have such a position which would be contrary to their construction
In this way we have proven that all straight lines e e eeof the plane s s ssa single such point corresponds to the oval line or a single interior point through which a single one of the lines e e eeThe corresponding tendon goes, and with this we have proven our theorem.

2. The polar surface of an interior point

The polar surface of an interior point B B BBlies entirely on the outer edge of the egg surface. On such B B BBpassing straight lines, for which B B BBThe polar surface has infinite points if the midpoint of the chord lying on the straight line is the polar surface. In all cases through B B BBThe polar surface has infinite points on the planes that intersect it; indeed, such a plane does not intersect the
The surface of an egg lies within an oval, and thus our claim is a consequence of a previously proven property of the interior points. It is possible that the surface of an egg possesses symmetrical regions or curves with respect to some interior points. Then, in the portion of space defined by the symmetrical regions/curves and the points in question, the polar surface of the point is part of the infinite plane. The cones defined by the boundary lines of the symmetrical regions and their centers of symmetry are the asymptotic cones of the polar surfaces of the point in question.
Due to the polar correspondence, there is also a relationship between planes and lines passing through an interior point. B B BBstraight line c c ccintersects the polar surface of the point B B BBat a single point P P PP(finite or infinite point). The polar surface of the point P P PPgoes through B B BBand has a well-defined tangent plane S at this point. This relationship is one-to-one.
The polar surface of an interior point has a well-defined tangent plane at all its points.
Be the point K K KKa point on the polar surface of the interior point B B BB. The line K B K B KBKBcuts the egg surface at the points A C A C ACAC; the planes passing through K B K B KBKBintersect the egg surface in egg lines, and the polar surface in through K K KKPlanar curves. The tangent of this curve at the point K K KKis through this point and through the intersection point D D DDthe tangents of the oval line at the points A C A C ACACdetermined. Point D D DDlies in the tangent planes of the points A A AAand B B BB, that is, on their line of intersection c c cc. In this way, the tangents of the lines lying on the polar surface and through lie K K KKCurves in the plane defined by the point K K KKand the line c. This plane is the tangent plane of the polar surface
A plane passing through an external point and not intersecting the egg surface is the tangent plane of a polar surface belonging to an internal point.
A plane passing through K K KKbut intersecting the surface of the egg cannot be a tangent plane to a polar surface. The plane passing through the point K K KKThe polar surfaces correspond to the points on the polar surface of the point. K K KK.
Let B B BBbe one of these points. The plane lines passing through the line K B K B KBKBintersect the egg-shaped surface in egg-shaped curves. The tangents of the polar curves of the point B B BBregarding these egg lines, which are in the by K K KKThe contact plane is - according to a property proven in the case of egg lines - located in the outer part of the egg lines and thus in the outer part of the egg surface.
Let B B BBa point on the polar surface of an external point K K KKThe polar surface of the point B B BBgoes through K K KKand has a well-defined tangent plane there. In this way, a corresponds to a K K KKplane of the line K B K B KBKB. These levels are defined by the points leading to the point K K KKThe adjoint congruence line is determined, and such a plane contains precisely that of the line. K B K B KBKBcorresponding lines of congruence. Level of contact.
We must therefore prove that in a through K K KKThe plane passing through and not intersecting the polar surface contains a single straight line of congruence.
Let s s ssa plane passing through K K KKthe egg surface but not intersecting it (Figure 19). Using the lines a of the plane, we construct the tangent planes of the egg surface and denote the line determined by their points of tangency by a 1 a 1 a_(1)a1. The points of tangency of those tangent planes which pass through the point K K KKThe straight lines are determined and lie on a curve. c c cc, the corresponding straight lines a 1 a 1 a_(1)a1link the pairs of points on the curve e e eeand thus give a one-to-one relationship of the curve e e eewith itself. Be k ( r ) k ( r ) k(r)k(r)a circle of fibene s s sswith the center K K KKand with the radius r r rrand we
Fig. 19
want to construct the tangent plane of the egg surface using all tangents to this circle. The corresponding lines a connect two points on the egg surface: one point lies on the surface O 1 O 1 O_(1)O1, the other one on O 2 O 2 O_(2)O2, where O 1 O 1 O_(1)O1and O 2 O 2 O_(2)O2the one from the curve e e eecertain parts of the egg surface. This fact is a consequence of the continuous change of the tangent planes. In this way, the lines intersect. a 1 a 1 a_(1)a1the cone shell K 1 K 1 K_(1)K1of the cone of contact on its between points K K KKand the curve e e eeLying parts at a single point A A AA(the cone of tangency is a convex surface) and thus each line a of the plane s corresponds to a single point. A A AAThe tangents of the circle k ( r ) k ( r ) k(r)k(r)corresponding points A A AAdescribe on K 1 K 1 K_(1)K1a continuous closed curve g ( r ) g ( r ) g(r)g(r). That straight line which defines the points of tangency of the with s s ssconnects parallel contact planes, intersects K 1 K 1 K_(1)K1at the point P P PP. The through K K KKThe curve corresponding to the straight line is the curve e e ee. As r r rrincreases, the curves describe g ( r ) g ( r ) g(r)g(r)the entire area K 1 K 1 K_(1)K1of the curve e e eeto the point P , K 1 P , K 1 P,K_(1)P,K1simply covering. In this way, for a certain r 0 r 0 r_(0)r0the curve g ( r 0 ) g ( r 0 ) g(r_(0))g(r0)through K K KKand the point K K KK, as a point on the curve g ( r 0 ) g ( r 0 ) g(r_(0))g(r0), corresponds to a single tangent b b bbof the circle k ( r 0 ) k ( r 0 ) k(r_(0))k(r0). According to the given construction, the line b 1 b 1 b_(1)b1- which are the points of contact of the b b bbincludes tangent planes - through K K KKThis b 1 b 1 b_(1)b1intersects the polar surface of the point K K KKin one point B B BBand the plane s s ssis the tangent plane of the polar surface of the point B B BBat the point K K KKThe construction is unambiguous, and thus we have proven our theorem.
  1. Received on November 25, 1959.
1959

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