Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 531-545.

A Family of Conics and Three Special Ruled Surfaces

Hans-Peter Schröcker

Institute for Architecture, University of Applied Arts Vienna Oskar-Kokoschka-Platz 2, A-1010 Wien, Austria e-mail: hans-peter.schroecker@uni-ak.ac.at

Abstract. In [5] the authors presented a family F = {c_{k} | k ∈ ℝ} of conics. The conics c_{k} are gained by offsetting from a given conic c_{0} with proportional distance functions kδ(t). We investigate certain properties of F and give the correct version of a result claimed in [5]: The distance function is unique (up to a constant factor) only if c_{0} is not a parabola. Furthermore we deal with the surfaces that are obtained by giving each conic c_{k} ∈ F the z-coordinate λk with a fixed real λ. We find special metric properties of these surfaces and show that they already appeared in other context. Keywords: conic, ruled surface, surface of conic sections, rational system of conics, refraction

1. Introduction

In this paper we deal with two geometric objects: A family of conics on the one hand and a class of special ruled surfaces of conic sections on the other hand. Both of them are closely related and properties of one object reflect as properties of the other.

Following an idea from [5] we derive a family F of conics through proportional offsetting from a given conic c_{0}: X(t) … x(t). The distance functions are given by λϱ(t)^{1/3} where ϱ(t) denotes the radius of curvature of c_{0} in X(t). This generation leads to a 3D-interpretation of F – the conics c_{λ} ∈ F (we will call them distance conics) may be regarded as the top view of the conics on a ruled surface. In Section 2 we use this connection to prove results on the envelope of the distance conics and on the uniqueness of the distance function if c_{0} is

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

not a parabola. In addition we find that the conics are projectively linked by the tangents of the evolute of c_{0}. Despite their connection to many geometric problems (some of them will appear in this paper), families of projectively linked conics have not been studied too extensively up to now.

In Section 3 we will investigate the ruled surfaces Φ_{e} and Φ_{h} of the elliptic and hyperbolic case. They are normal surfaces of two quadrics of revolution along a common plane section parallel to the axes. In addition to their attractive shape and ruled character (which makes them interesting for architectural usage, compare [5]), they have geometric applications as well. Only recently they appeared in a paper on the geometry of refraction ([4]).

Section 4 deals with the surface Φ_{p} of the parabolic case. It is not the normal surface of a quadric along a plane section but has special properties concerning the line of striction. Φ_{p} is an example of Steiner’s Roman surface with a two parameter manifold of parabolas on it. It is a conoid and affinely equivalent to a surface studied in [10].

Returning to families of conics we finally find that the distance function is not unique in the parabolic case. There exists a one parameter family of non-proportional distance functions that can be used for the generation of families of distance conics as desribed above.

2. A family F of conics and a ruled surface Φ

2.1. Creating conics by proportional offsetting

Let c_{0} : X(t) … x(t) be a parametrized conic in the Euclidean plane 𝔼^{2}. Being given an arbitrary distance function δ(t) of the curve parameter we can construct the curve at distance δ(t)

c_{δ(t)}: Y (t) … y(t) = x(t) + δ(t)n(t).

In this formula n(t) denotes the unit normal vector of c_{0} in X(t). If δ(t) ≡ d is constant, c_{δ}(t) is the ordinary offset curve at distance d.

Of course, any plane curve can be parametrized (at least locally) in this way. One has to make further assumptions on δ(t) in order to get an object of geometric interest. In [5] the authors asked for a function δ(t) with the property that for any real k the curve at distance kδ(t) is a conic. There, they claimed the following result (compare Figure 1):^{1}

Being given an arbitrary conic c_{0} there exists (up to a constant factor) exactly one function δ(t) such that for any real k the curve c_{k} at distance kδ(t) is a conic as well. If ϱ(t) is the radius of curvature of c_{0} in X(t) one may use δ(t) = ϱ(t)^{1/3}.

They provided, however, no detailed proof for the uniqueness of the distance function. Only an outline for the elliptic case was given. Their proof involved the usage of a computer algebra system and was too long to be published. Apparently the authors missed the fact that the offset function is not unique if c_{0} is a parabola.

We will give a simple proof for the elliptic and hyperbolic case in Subsection 2.3. In Subsection 4.3 we will investigate the parabolic case and describe a geometric method of constructing all distance functions with the requested property.

1We used the software provided in [3] to produce the pictures in this paper.

Figure 1: A single distance conic c_{k} of c_{0} (left) and the family F of distance conics (right).

Before doing so, we investigate the family F := {c_{k} | k ∈ ℝ} of distance conics to an arbitrary base conic c_{0}. The distance function used to generate c_{k} shall be kϱ(t)^{1/3} as proposed in [5]. For the time being only the members of this family F will be called distance conics. Later on (in Subsection 4.3), we will deal with other families of conics gained through proportional offsetting as well.

A conic’s evolute has as many real points at infinity as the conic itself. Each point at infinity corresponds to a pole of the function ϱ(t). Hence, all regular distance conics are of the same type as c_{0}. In general, they do not belong to a pencil of conics. An exception is the circular case where F is a set of concentric circles. We will exclude this from our considerations until Subsection 4.3.

If c_{0} is not a parabola we can easily express the semiaxes a_{k}, b_{k} of c_{k} in terms of the semiaxes a, b of c_{0} ([5]):

a_{k} = a + \frac{kb}{w},  b_{k} = b + ε\frac{ka}{w},  (1)

where w = (ab)^{1/3} and ε = ±1 in the elliptic and hyperbolic case, respectively. (1) shows that the two distance conics corresponding to

k_{1} := −\frac{aw}{b} and k_{2} := −ε\frac{bw}{a}

degenerate to line segments. In the elliptic case these segments are finite, in the hyperbolic case they are infinite.

Now to the parabolic case. In an appropriate Cartesian coordinate system c_{0} can always be described by the equation y = ax^{2}. Then the conic section c_{k} is given by y = p_{k}x^{2} + q_{k}, where

p_{k} = \frac{a}{(1 + kr^{2})^{2}} , q_{k} = −\frac{k}{r} and r = \sqrt[3]{2a}.  (2)

Here, the single degenerating distance conic corresponds to k = −r^{−2}. It is easy to see that in any of the three cases c_{0} is the only conic that yields F as its family of distance conics. I.e., different base conics have different families of distance conics.

On the right hand side of Figure 1 one can see a conic c_{0}, its evolute e and the corresponding family F of distance conics. Apparently e is the envelope of all conics c_{k} and we will soon give a proof for this. But first we need to know a little more about the family F of distance conics and an interpretation in 3-space. We could use the formulas (1) and (2) to find the envelope of F by standard methods. Our proof, however, will turn out to be much easier.

2.2. Associated ruled surfaces of conic sections

The possibility of creating a family of conic sections by offsetting with proportional distance functions is the key for a 3D-interpretation of F. We choose a Cartesian coordinate system {O; x, y, z} where c_{0} is defined by z = 0 and

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, \frac{x^{2}}{a^{2}} − \frac{y^{2}}{b^{2}} = 1 or y = ax^{2}

in the elliptic, hyperbolic or parabolic case, respectively. By assigning the z-coordinate λk (λ ∈ ℝ \ {0}) to each distance conic we create a surface Φ of conic sections. In [5] this surface has been presented and visualized for all three types of base conics. But still, Φ has many remarkable properties that were not mentioned there and will be contents of this paper. Images of Φ for the elliptic and hyperbolic case may be found in Section 3. The surface of parabolic sections is displayed in Section 4.

Obiously, by the way it has been defined, Φ is not only a surface of conics but a ruled surface as well. The top view of the generating lines just yields the normals of c_{0}. Thus, according to [1], Φ is algebraic of order n ≤ 4. If c_{0} is an ellipse or a hyperbola it has two double lines

d_{1} … x = 0, z = λk_{1} and d_{2} … y = 0, z = λk_{2}

that stem from the two degenerating conics. As a ruled surface of order n ≤ 3 never has two double lines ([6]) we get n = 4 in these cases. In the parabolic case we have n = 3 as we shall see in Section 4. Now we prove the theorem on the hull curve of F making essential use of the presented 3D-interpretation (compare Figure 1):

Theorem 1. The evolute e of c_{0} is an envelope of the family F of distance conics.

Proof. Projecting Φ orthogonally on the [x, y]-plane yields a contour line l_{1} ⊂ Φ. Its projection l′_{1} on [x, y] consists of the hull curve e of the projected generating lines of Φ plus possible components that result from singular surface points. e is the evolute of c_{0} and – at the same time – the envelope of the projected conics of Φ, i.e., the conics of F.

Theorem 1 is only a special case of a more general theorem.

Theorem 2. Being given a differentiable plane curve d: Y (t) … y(t) and an arbitrary function ε(t) we construct a set {d_{k} | k ∈ ℝ} of curves at distance kε(t). Then the evolute of d is envelope of all curves d_{k}.

Proof. The proof is completely analogous to the proof of Theorem 1. The relevant property of the curves d_{k} is the possibility of building a ruled surface in space by assigning the zcoordinate λk to d_{k}.

2.3. Uniqueness of the distance function in the elliptic and hyperbolic case

We first present some basic facts on the class curve or ruled surface generated by two projectively linked conic sections (compare [6, 7]). Let c and d be two conics in a common plane that are projectively linked by the map π: c → d. We regard the set c^{∗} of lines passing through corresponding points C ∈ c and π(C). If c^{∗} happens to be a pencil of lines we require that its vertex S lies in c ∩ d.^{2} Now the following statements are equivalent:

• c^{∗} is a rational curve of class γ (γ = 1, 2, 3, 4).

• There exist exactly 4 − γ fixpoints of π.

• There exists a (5 − γ)-parameter manifold of conics that are projectively linked by the

tangents of c^{∗} and, thus, may be used to generate c^{∗} instead of c or d. If c and d are conics in 3-space they do not generate a class curve but a ruled surface Ψ. We can, however, state three equivalent points that read very similar:

• Ψ is an algebraic surface of order γ (γ = 2, 3, 4).

• There exist exactly 4 − γ fixpoints of π.

• On Ψ we find a (5 − γ)-parameter manifold of conics that are projectively linked by

the generators of Ψ and, thus, may be used to generate Ψ instead of c and d. It is noteworthy that any ruled surface of conic sections can be generated by two projectively linked conics. This was stated in [1] for the first time. For the proof of the following theorem it will be important to observe that any two conics on a ruled surface of conic sections are projectively linked by the generators.

Theorem 3. Being given an arbitrary ellipse or hyperbola c_{0} there exists (up to a constant factor) exactly one function δ(t) such that for any real k the general offset curve c_{k} with distance function δ_{k}(t) := kδ(t) is a conic as well. If ϱ(t) is the radius of curvature of c_{0} in X(t) one may use δ(t) = ϱ(t)^{1/3}.

Proof. [5] garantuees that the curve c_{k} at distance kϱ(t)^{1/3} is a conic again. In order to prove the uniqueness we take another function δ̃(t) that, too, for any k ∈ ℝ yields a conic c̃_{k} as curve at distance kδ̃(t). Then we associate ruled surfaces Φ and Φ̃ with the families F := {c_{k} | k ∈ ℝ} and F̃ := {c̃_{k} | k ∈ ℝ}, respectively. The conics on Φ and Φ̃ are now projectively linked by the generators of these surfaces. Thus, each two conics from F and/or F̃ are projectively linked by the normals of c_{0}. According to Theorem 2 any two of these conics generate the evolute e of the base conic c_{0}.

A conic c̃ ∈ F̃ \ F can now be used to construct a two parameter manifold of conics with this property. Any conic c^{∗} ∈ F is projectively linked to c̃ and, thus, determines a ruled surface Φ^{∗}. The top view of the conics on Φ^{∗} yields a one parameter family of generating conics different from F. This is, however, not possible in the elliptic or hyperbolic case as e is of class 4. Therefore we get F = F̃ and δ̃(t) = kϱ(t)^{1/3} with some constant k ∈ ℝ.

The proof of Theorem 3 is not valid in the parabolic or circular case. The evolutes of these curves are of class 3 or 1, respectively, and the existence of a two parameter set of generating conics is no contradiction. We will solve the problem for these cases in Subsection 4.3. The following corollary follows from the proof of Theorem 3 and is valid in any case:

2If S ∉ c ∩ d we get a case that is of no relevance in this context. It will, however, show up again in Subsection 4.3.

Corollary 1. The conics of F are projectively linked by the normals of c_{0}.

For further investigations we need parameter representations of Φ. We will therefore treat the three cases (elliptic, hyperbolic and parabolic) separately. In order to distinguish between them, we will henceforth denote the ruled surface of conic sections by Φ_{e} (elliptic), Φ_{h} (hyperbolic) and Φ_{p} (parabolic).

3. The surfaces Φ_{e} and Φ_{h} of elliptic and hyperbolic sections

3.1. The normal surface through the base conic

From (1) it follows that Φ_{e} can be parametrized according to

Φ_{e}: X(t, u) … X(t, u) = (1 − u)d_{1}(t) + ud_{2}(t).  (3)

In this formula d_{1} and d_{2} are parameter representations of the double lines d_{1} and d_{2} of Φ_{e}:

d_{1}: D_{1}(t) … d_{1}(t) = \frac{1}{b}(0, (b^{2} − a^{2}) sin t, −λaw)^{T} ,

d_{2}: D_{2}(t) … d_{2}(t) = \frac{1}{a}((a^{2} − b^{2}) cos t, 0, −λbw)^{T} .

The u-parameter lines of (3) are the generators g(t) of Φ_{e}, the t-parameter lines are the conics on Φ_{e}. They are situated in the planes through the line at infinity of the [x, y]-plane which is a double line of the surface. The base conic c_{0} lies in [x, y] and belongs to u = a^{2}(a^{2} − b^{2})^{−1}. Analogous formulas and statements are true for Φ_{h}. We will, however, not display them here. They can be obtained by replacing the circular functions sin(t) and cos(t) by their hyperbolic relatives sinh(t) and cosh(t), respectively, and changing the sign of certain terms. In Figure 2 both surfaces are to be seen.

Figure 2: The ruled surfaces Φ_{e} and Φ_{h}.

c_{0} is a special conic in the family F of distance conics in the way that it is characteristic of F. No other conic produces the same family of distance conics. So we might expect certain special properties of c_{0} as conic on Φ_{e} as well. The investigations to follow will show that Φ_{e} has already been studied in earlier works and in other context. Besides, it will be the basis for a very important application of Φ_{e} in geometrical optics.

By the scaling transformation x ↦ ab^{−1}x or y ↦ ba^{−1}y we can map D_{1}(t) and D_{2}(t) to points of constant distance

d = \frac{| a^{2} − b^{2} |}{ab}\sqrt{λ^{2}w^{2} + a^{2}} or d = \frac{| a^{2} − b^{2} |}{ab}\sqrt{λ^{2}w^{2} + b^{2}},

respectively. Thus, Φ_{e} may be transformed to a ruled surface Φ_{e}^{∗} that consists of all straight lines that intersect two perpendicular straight lines d_{1} and d_{2} in points of constant distance. This surface was presented in [2] where it was called Reiterfläche (rider surface). Note that there exists no family of distance conics that can be associated to the rider surface. In a top view the conics of Φ_{e}^{∗} envelope an astroid that can never be the evolute of a conic. Now we want to solve the following task: Find a point S ∈ z and a conic c: X(t) … X(t, u_{0}) of Φ_{e} such that each generating line g(t) of Φ_{e} is

1. perpendicular to the straight line [S, X(t)].

2. perpendicular to the tangent of c in X(t). In other words: Find a cone Γ of second order with apex on z such that Φ_{e} is the normal surface of Γ along the plane section c. It is easy to answer this question algebraically. There exists a unique point S and a unique conic c on Φ_{p} satisfying the first condition:

S(0, 0, w^{2}λ^{−1})^{T} , u_{0} = \frac{a^{2}}{a^{2} − b^{2}} .

But, as c is just the base conic c_{0}, it meets the second condition as well. Thus we have

Theorem 4. Φ_{e} is the normal surface along c_{0} of the cone Γ with center S.

Surfaces of that kind have been studied in a more general context in [6] and in some earlier publications as well. However, because of its symmetry (the plane section c_{0} shares two planes of symmetry with Γ), our case is rather special. E. g., the following remarkable theorem (it follows from some considerations in [6]) is true:

The line of striction on Φ_{e} is the contour with respect to the projection center S.

3.2. Two quadrics of revolution through the base conic

Of course, Φ_{e} is not only normal surface of Γ but of any quadric being tangent to Γ along c_{0}. These quadrics belong to a pencil Q. The implicit equation of any member of Q can be written as

αG(x, y, z) + βH(z) = 0,

where

G(x, y, z) := \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} − \frac{(w^{2} − λz)^{2}}{w^{4}} and H(z) := z^{2}

Figure 3: Φ_{e}, its normal cone Γ and the ellipsoids of revolution Q_{1}, Q_{2}.

are the implicit equations of Γ and [x, y] as double plane, respectively. α: β is a homogeneous parameter of the pencil Q. Two quadrics Q_{1} and Q_{2} are of special interest (compare Subsection 3.3). They belong to

α : β = a^{2}w^{4} : w^{4} + a^{2}λ^{2} and α : β = b^{2}w^{4} : w^{4} + b^{2}λ^{2}

and are ellipsoids of revolution with axes d_{1} and d_{2}, respectively.

Corollary 2. Φ_{e} is the normal surface of two ellipsoids of revolution with axes d_{1} and d_{2}, respectively, along c_{0}.

Figure 3 shows Φ_{e} together with its normal cone Γ and two quarters of the ellipsoids of revolution Q_{1} and Q_{2}.

3.3. Φ_{e}, Φ_{h} and refraction on a plane

Besides the architectural suggestions made in [5] both surfaces, Φ_{e} and Φ_{h}, have an important application in the geometric theory of refraction that shall be explained here (compare [4]).^{3} We will shortly outline some basics on refraction in the plane case.

Let σ be a plane (the refracting plane) and r a positive real (the index of refraction). A straight line b_{1} intersecting σ in a point B is now refracted to a straight line b_{2} = R(b_{1}) according to Snell’s law

sin α_{1} = r sin α_{2}  (4)

where α_{i} denotes the angle between b_{i} and the normal n of σ (Figure 4). Now we choose a point E (the eye point) and reflect all rays through E on σ. It is not hard to prove that the congruence of all reflected rays is the normal congruence of a quadric of revolution Q. To be

3The author would like to express his gratitude to M. Husty for pointing out this connection.

Figure 4: Snell’s law and the “refrax” R(X) of a space point X.

more precise, Q is an ellipsoid if r < 1 and a hyperboloid if r > 1.^{4} In both cases the axis of revolution is the normal of σ through E and this point a focal point of Q. The half length of the axes of Q can easily be expressed in terms of e (the distance of E and σ) and r. Now we can define a map

R : 𝔼^{3} → σ, X ↦ R(X) = Y

that assigns to each point X ∈ 𝔼^{3} the point Y ∈ σ with R([E, Y]) = [X, Y], i.e., the point we practically try to look at when we want to see X through the refracting plane σ (Figure 4).^{5} This map is important in computer graphics for the direct and efficient computation of refraction images. Furthermore, it can be applied to create curved perspectives as well.

The counter image Ψ := R^{−1}(s) of a straight line s ⊂ σ is a ruled surface that can be used to solve certain problems of theoretical and practical relevance. E. g., it helps to answer questions on the R-image of an algebraic curve a of order n (in general R(a) is an algebraic curve of order 4n) and to handle problems with standard filling algorithms for polygons in computer graphics (the R-image of a polygon P may have up to two overlappings; the criterion for this is the number of generators of Ψ that intersect P).

Probably you already guessed that Ψ = Φ_{e} or Ψ = Φ_{h} and, of course, you are right. For a change we will deal with the hyperbolic case in this subsection. We will use a Euclidean coordinate system where σ is the [x, y]-plane, E lies on the z-axis and the straight line s is described by z = 0, y = s_{y}. Then all reflected rays are perpendicular to a hyperboloid H of revolution

H: H(u, v) … H(u, v) = (\begin{matrix}a cosh u sin v \\ a cosh u cos v \\ b sinh u \end{matrix}) .  (5)

Now we compute the normals of H that intersect s. The characteristic condition for this is

cosh u cos v = \frac{as_{y}}{a^{2} − b^{2}} .

4If r = 1 the refraction is actually just a reflection and Q degenerates.

5Actually R is not well defined by this condition as there may be up to four real solutions. There exists, however, exactly one point that is relevant for practical purposes and it can easily be characterized ([4]).

Figure 5: The counter image Φ_{h} = R^{−1}(s) of a straight line s.

Together with (5) this yields

The counter image Ψ of a straight line s is the normal surface of a hyperboloid H of revolution along a plane section parallel to the axis of H. Hence, it is an example of the ruled surface Φ_{h}.

Figure 5 shows the situation. The double lines of Φ_{h} are the z-axis and the straight line s, the eye point E is focal point of H. For practical applications only one sheet of Φ_{h} is relevant. But taking into account that, from the mathematical point of view, equation (4) has two solutions in [−\frac{π}{2}, \frac{π}{2}] we get the whole surface Φ_{h}.

4. The surface Φ_{p} of parabolic sections

4.1. Basic properties

The parameter representation of the surface Φ_{p} of parabolic sections given by [5] can easily be derived from (2). With the abbreviation r = (2a)^{1/3} it reads

Φ_{p}: X(t, u) … X(t, u) = (\begin{matrix}t + tur^{2} \\ at^{2} − ur^{−1} \\ λu \end{matrix}) .  (6)

The t-lines are parabolas, the u-lines are generators g(t) of the surface. The base parabola c_{0} belongs to u = 0. Φ_{p} has the double line

d … x = 0, z = −λr^{−2}

and is symmetric with respect to [y, z]. The [y, z]-plane intersects Φ_{p} in the straight line

s … x = 0, ry + z = 0

and the double line d. It is easy to see that Φ_{p} is a conoidal surface. All its generating lines are parallel to the plane ry + z = 0. The parametrization (6) has the characteristic shape of the Weierstrass-representation of Steiner’s Roman Surface ([8, 11]): The homogeneous coordinates x_{0} :x_{1} :x_{2} :x_{3} of the surface points are quadratic polynomials of the two parameters t and u. In our case, already the Euclidean coordinates fulfil this condition. The Roman Surface and any of its tangent planes have two conics in common. Thus, there exists a two parameter manifold of conics on Φ_{p}. Its algebraic order must therefore be three.^{6}

A surface very similar to Φ_{p} has already been investigated more than 30 years ago in [10]. Using an affine coordinate system with origin S := s ∩ d and coordinate axes parallel to x, y and s we can parametrize Φ_{p} as

Φ_{p}^{∗}: X^{∗}(t, u) … X^{∗}(t, u) = (\begin{matrix}tu \\ t^{2} \\ u \end{matrix}) .  (7)

This is – up to a permutation of the coordinates – identical to a parameter representation that was used in [10] for the investigation of a certain ruled surface of order three. There, the author studied affine properties of Ψ (that, of course, hold for Φ_{p} as well). So he could afford choosing the normalized parameter representation (7). Φ_{p} has, however, an interesting metric property that is lost by transforming it to Ψ.

4.2. The line of striction

Some computation following the example of Section 3 shows that Φ_{p} is not the normal surface of any cone of second order along a plane section. To be more detailed: The conics (parabolas) on Φ_{p} are defined by relations of the kind u = αt + β. Evaluating the condition that SX(t) and g(t) are perpendicular immediately yields α = 0 and β = −r(4aλ)^{−1} which defines a unique parabola c_{p} ⊂ Φ_{p}. The normals of Φ_{p} along c_{p} are, however, not the generators of a cone or cylinder.

Instead we will have a closer look at the line of striction l: L(t) … l(t) of Φ_{p}. Φ_{p} is a conoid, so l is the contour for the normal projection on [x, s]. As a consequence, all tangent planes of Φ_{p} along l are parallel to v := (0, r, 1)^{T}. Their hull surface is a cylinder that will be denoted by ζ_{s} in the following. The line of striction can be computed from (6). It has the polynomial representation

l: L(t) … l(t) = \frac{1}{2a(1 + r^{2})} (\begin{matrix}−4a^{3}t^{3} \\ 2a^{2}t^{2}(r^{2} + 3) + r^{2} + 1 \\ −4a^{2}rt^{2} − 2a − r \end{matrix}) .

6A ruled surface of conics is of order three iff there is a two parameter family of conics on it ([6]). Of course, all conics on Φ_{p} are parabolas.

Figure 6: The surface Φ_{p} and the cylinder ζ_{s}.

Each generating line of ζ_{s} intersects Φ_{p} in three points. Two of them coincide in a point on l, the third one lies on a curve k with parameter representation

k: K(t) … k(t) = \frac{1}{4ar(r + 2a)} (\begin{matrix}2a^{3}t^{3}r^{2} \\ r^{2}[2r^{2}(2a^{2}t^{2} + 1) + 3a^{2}t^{2} + 2] \\ −2a(2r^{2} − a^{2}t^{2} + 2) \end{matrix})

It is easy to verify that l and k are plane curves. As we have l̇(0) = k̇(0) = o, the points L(0) and K(0) are cusps of l and k. Thus, these curves are cubic parabolas (evolutes of parabolas). As v is orthogonal to s we can state (compare Figure 6)

Theorem 5. The line of striction l of Φ_{p} is a plane cubic parabola. It is the contour for the normal projection on the plane [x, s]. The projection rays through l intersect Φ_{p} in another plane cubic parabola k.

4.3. Distance conics to parabola and circle

We already mentioned the result stated in [5]: The general offset curve of a base conic c_{0} is a conic if the distance function δ(t) is of the shape δ(t) = kϱ^{1/3}(t) where ϱ(t) is the radius of curvature at X(t) and k an arbitrary real. In Theorem 3 we gave a proof for the uniqueness (up to a constant factor) of the distance function if c_{0} is not a parabola. This proof would not work in the parabolic case, however. In fact, the distance function is not unique as we will see. We already know some basic facts (compare Theorem 2 and the proof of Theorem 3):

• There exists a two parameter manifold P of parabolas that are projectively linked by the tangents of the evolute e of c_{0}.^{7}

• Only these parabolas can be members of families of distance conics with proportional distance functions.

• All parabolas of P can be parametrized by substituting u = αt+β in (6) and cancelling

the z-coordinate. In other words: They are just the top view of the parabolas on Φ_{p}. Because of the last point, any parabola p = p(α, β) from P corresponds to a straight line p^{∗} … u = αt + β in the [t, u]-parameter plane. The straight lines t = const. belong to the generators of Φ_{p}. Being given an arbitrary parabola p(α, β) and a real k ∈ ℝ we build the linear combination

c_{k}: X_{k}(t) … (1 − k)X(t, 0) + kX(t, αt + β)  (8)

(8) is the parameter representation of a parabola c_{k} of P. Furthermore, c_{k} is the curve at distance δ(t) from c_{0} where

δ(t) = |X(t, αt + β) − X(t, 0)| .  (9)

Thus, we found a one parameter set 𝔽 of families of conics with the requested property (see Figure 7). Because of Theorem 2 the hull curve of any of these families is the evolute e of c_{0} again. There exists exactly one parameter value t_{0} with X(t_{0}, 0) = X(t_{0}, αt_{0} + β), i.e., t_{0} yields the same point P on c_{0} and p(α, β). All parabolas of the family defined by c_{0} and p(α, β) pass through P. Two families F_{1}, F_{2} ∈ 𝔽 have only the base parabola c_{0} in common. Thus, 𝔽 is maximal and we get

Theorem 6. Being given a parabola c_{0} there exists a one parameter family of distance functions (9) that can be used to create different families of distance conics (8).

The parabolas of a family F_{1} ∈ 𝔽 correspond to a family F_{1}^{∗} of parabolas on the surface Φ_{p}. There, the parabolas are located in planes passing through a fixed point P ∈ c_{0} and being tangent to Φ_{p}. We denote this set of tangent planes by τ(P). They envelope a quadratic cone Γ and all generators of Φ_{p} are tangents of Γ. As Φ_{p} is a conoid, the planes of τ(P) intersect any two generators of Φ_{p} in points corresponding in an affine transformation.

In the [t, u]-parameter plane the families of 𝔽 correspond to the pencils of lines with vertices on the straight line s_{0} … t = 0 that itself corresponds to the base parabola c_{0}. The distance function δ(t) = ϱ(t)^{1/3} yields the pencil of lines parallel to s_{0}. This distance function – the choice of [5] – is characterized by the existence of a singular distance conic – not only in the parabolic but also in the elliptic and hyperbolic case.

Theorem 6 is only possible because a parabola’s evolute is of class 3 only. But there is another conic with an evolute of class γ ≤ 4: the circle. Its evolute e degenerates to a single point and is of class 1. The proof of Theorem 3 is, thus, not valid for a circle.^{8} To complete

7Certain parabolas may degenerate to straight lines. This has, however, no effects on the reasoning to follow.

8Remember that we excluded this case at the very beginning of this paper.

Figure 7: A parabola c_{0}, its evolute e and a set of proportional offset parabolas.

the picture we will investigate this case right here. This can be done by straightforward computation. We parametrize c_{0} according to

c_{0}: X(t) … x(t) = (\begin{matrix}r cos t \\ r sin t \end{matrix}) .  (10)

As the apex O of the degenerated evolute e does not lie on c_{0} there exists a three parameter manifold C of conics that are projectively linked by the tangents of e.^{9} An arbitrary conic c ∈ C can be parametrized according to

c: Y (t) … y(t) = \frac{1}{r(a cos t + b sin t) + c} (\begin{matrix}r cos t \\ r sin t \end{matrix})

with three reals a, b and c. Now we solve the vector equation y(t) = x(t) + kr^{−1}f(t)x(t) for the function f(t) and find

f(t) = −\frac{r^{2}(a cos t + b sin t) + r(c − 1)}{kr(a cos t + b sin t) + kc}.

Varying k yields a proportional function f(t) only if a = b = 0 and we get the trivial family of concentric circles as the only solution to the problem. Theorem 3 is therefore valid for the circular case as well.

5. Conclusion and future work

In this paper we solved the question of finding all families of proportional distance conics to a given conic. In contrast to claims in [5] there exists more than one family of that kind in the parabolic case. These families are closely linked to certain ruled surfaces of conic sections that have remarkable geometric properties and applications. Perhaps there will be found more then we gave in the above text. Of special interest seems the relation of Φ_{e} and Φ_{h} to the surfaces presented in [2].

9See [7]; all conics may be found as the top view of the plane sections of an arbitrary cone of revolution with axis z.

Throughout the text we dealt with families of conic sections that are projectively linked by straight lines (the tangents of a rational curve of class γ ≤ 4 or the generators of a ruled surface of conic sections). Many results (e. g., those of Subsections 2.3 and 4.3) on these families can be obtained by investigating linear combinations of two rational parametrizations that realize the projective mapping by identical parameter values.

It is possible to combine more than two parameter representations and create moredimensional rational systems of conics (see [7]). They have the structure of a projectiv space and can be applied to many geometric problems. In contrast to the theory of pencils of conics there exists no theory of rational systems of conics yet. Hopefully, this paper has shown their usefullness and will promote their further investigation.

References

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Received February 6, 2000; revised version September 20, 2000