Startseite Mathematik Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space
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Pedal and negative pedal surfaces of framed curves in the Euclidean 3-space

  • Kaixin Yao und Donghe Pei EMAIL logo
Veröffentlicht/Copyright: 13. November 2025
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

We define pedal and negative pedal surfaces of framed curves in the Euclidean 3-space and find that the loci of singularities of them are pedal and negative pedal curves, respectively. Moreover, we give sufficient conditions that pedal and negative pedal surfaces to be framed base surfaces. We also give a sufficient condition that negative pedal curves to be framed base curves.

Keywords: pedal surface; negative pedal surface; negative pedal curve; framed curve; framed surface
MSC 2020: 53A05; 53A04; 57R45

1 Introduction

Curves or surfaces generated by curves are important branches in differential geometry. Some properties of the original curve can be reflected by the new curve or surface. For example, given a smooth regular curve with a vertex in the Euclidean plane, its evolute is singular at the point corresponding to the vertex of the original curve. There are also relations between curves and surfaces generated by the same curve. For a spatial curve, the singularity locus of its focal surface coincides with its evolute [1] (Figure 1).

Figure 1: A focal surface (mesh) and an evolute (red).
Figure 1:

A focal surface (mesh) and an evolute (red).

People first considered that original curves are regular. In fact, there is a limitation. With the development of Legendre curves, people can deal with this kind of curves, which may have singularities, similarly to regular curves [2]. A Legendre curve can be determined by its invariants uniquely up to rigid motions. For a Legendre curve (Legendre immersion), its base curve is called a frontal (front). Lifting the dimension of the space, Honda and Takahashi defined framed curves in the Euclidean n-space [3]. There is more than one way to frame a curve and different frames can be used to research its properties [4]. One can find an adapted frame to make some questions easier. Honda and Takahashi defined the Frenet type frame along a framed base curve in the Euclidean 3-space [1],5]. So far, framed curves with Frenet type frames can be regarded as the generalization of Frenet curves.

Roulettes and pedal curves play an important role in the study of road-wheel pairs. Pedal curves have connections with moving wheels [6]. In astronomy, Kepler discovered the trace of a moving planet is an ellipse. The ellipse is called Kepler’s ellipse. Hamilton claimed that the pedal curve of Kepler’s ellipse relative to the focus is a circle [7]. That is to say the ellipse is the negative pedal curve of the circle. Later, pedal curves in different spaces have been studied by many researchers. People were first concerned about pedal curves of regular curves. They found the pedal curve is singular if the original curve has inflections or the pedal point lies on the original curve. Later, Nishimura defined pedal curves in n-dimensional sphere and studied normal forms for singularities of them [8],9]. Using the theory of Legendre curves, people defined pedal curves of fronts and frontals in the Euclidean plane [10],11]. It contains the case of regular plane curves. However, pedal curves of frontals may be not frontals. Tuncer et al. gave a sufficient condition when the pedal curve of a front is a frontal. We studied the pedal curves of framed curves with Frenet type frames in the Euclidean 3-space [12]. Li and the second author of this paper researched pedal-contrapedal curve pairs of fronts in the unit sphere and discussed their relationships with the involute-evolute curve pairs [13]. As for other spaces, such as Minkowski space and its submanifolds, pedal curves were defined and studied [14], [15], [16]. If the orthogonal projection is changed to the slant projection, one can get pedaloids in plane. There are some results about pedaloids in the Euclidean plane and the Lorentz plane [17], [18], [19]. Pedal curves in a plane have applications in physics. One can use pedal curves to generate coordinates, called pedal coordinates, to deal with some mechanical problems [20], [21], [22].

Naturally, there is an inverse problem: if we know a curve and a fixed point, how can we get a new curve such that its pedal curve relative to the fixed point is the given curve? The new curve is called the negative pedal curve or the primitive. Arnol’d researched the primitive of a hypersurface [23]. It is the envelope of the normal hyperplanes. Izumiya and Takeuchi introduced the primitivoids of plane curves, which are generalizations of primitives [24]. For n dimensional frontals in the (n + 1) dimensional Euclidean space, Janeczko and Nishimura defined their negative pedals [25]. This is the generalization of the case of Arnol’d.

In the Euclidean plane, the pedal curve is the envelope of a family of circles and the primitive is the envelope of a family of lines. Taking the pedal curve and taking the primitive can be regarded as inverse processes. We consider envelopes of a family of spheres and a family of planes in the Euclidean 3-space, call them the pedal surface and the negative pedal surface, respectively. Previously, people usually studied pedal surfaces of surfaces [26]. It is a transformation of the same dimension. We change the correspondence of dimension. Based on our previous work [12], we research pedal surfaces and negative pedal surfaces.

In the present paper, we define pedal surfaces, negative pedal surfaces and negative pedal curves of framed curves. The singular locus of the pedal surface is the pedal curve and the pedal point. For the singular locus of the negative pedal surface, if we can establish the Frenet type frame along it and take its pedal curve, then it is the original curve. So we call it the negative pedal curve. We get sufficient conditions when the pedal surface and the negative pedal surface are framed base surfaces. Finally, we give two examples to show connections among pedal curves, negative pedal curves, pedal surfaces and negative pedal surfaces.

All maps and manifolds considered here are differentiable of class C ∞ .

2 Preliminaries

Let R 3 be the Euclidean 3-space with the inner product ⋅, the vector product × and the norm ‖⋅‖. We denote Δ = {( a , b ) ∈ S 2 × S 2| a ⋅ b = 0}.

Definition 2.1

([3]). I is an interval of R . ( γ , ν 1 , ν 2 ) : I → R 3 × Δ is called a framed curve if γ′(t) ⋅ ν 1(t) = γ′(t) ⋅ ν 2(t) = 0 for all t ∈ I. We call γ : I → R 3 a framed base curve if there exists ( ν 1, ν 2) : I → Δ such that (γ, ν 1, ν 2) is a framed curve.

For a framed curve (γ, ν 1, ν 2), define μ (t) = ν 1(t) ×  ν 2(t). Then { ν 1(t), ν 2(t), μ (t)} is a moving frame along γ(t). The Frenet type formulas are

ν 1 ′ ( t ) ν 2 ′ ( t ) μ ′ ( t ) = 0 l ( t ) m ( t ) − l ( t ) 0 n ( t ) − m ( t ) − n ( t ) 0 ν 1 ( t ) ν 2 ( t ) μ ( t ) , γ ′ ( t ) = α ( t ) μ ( t ) ,

where

l ( t ) = ν 1 ′ ( t ) ⋅ ν 2 ( t ) , m ( t ) = ν 1 ′ ( t ) ⋅ μ ( t ) , n ( t ) = ν 2 ′ ( t ) ⋅ μ ( t ) , α ( t ) = γ ′ ( t ) ⋅ μ ( t ) .

The map ( l , m , n , α ) : I → R 4 is called the curvature of the framed curve (γ, ν 1, ν 2). It is clear that γ is singular at t 0 if and only if α(t 0) = 0. We always suppose singularities of γ are discrete.

By the above definition, we know the linear combination of ν 1(t) and ν 2(t) is orthogonal to γ′(t). If (m(t), n(t)) ≠ (0, 0) for all t ∈ I, we take

n 1 ( t ) n 2 ( t ) = 1 m 2 ( t ) + n 2 ( t ) m ( t ) n ( t ) − n ( t ) m ( t ) ν 1 ( t ) ν 2 ( t ) .

Then (γ, n 1, n 2) is also a framed curve and n 1(t) ×  n 2(t) = μ (t). { n 1(t), n 2(t), μ (t)} is called the Frenet type frame along γ(t). The Frenet type formulas are

n 1 ′ ( t ) n 2 ′ ( t ) μ ′ ( t ) = 0 L ( t ) M ( t ) − L ( t ) 0 0 − M ( t ) 0 0 n 1 ( t ) n 2 ( t ) μ ( t ) , γ ′ ( t ) = α ( t ) μ ( t ) ,

where

L ( t ) = m ( t ) n ′ ( t ) − m ′ ( t ) n ( t ) + l ( t ) ( m 2 ( t ) + n 2 ( t ) ) m 2 ( t ) + n 2 ( t ) , M ( t ) = m 2 ( t ) + n 2 ( t ) .

The map ( L , M , 0 , α ) : I → R 4 is the curvature of (γ, n 1, n 2) [1],5]. The n 1, n 2, μ direction are called the principal normal direction, the binormal direction and the tangent direction of γ. In the present paper, we always use (γ, n 1, n 2) to represent a framed curve with the Frenet type frame { n 1, n 2, μ } and the curvature (L, M, 0, α). The surface which consists of all tangent lines of γ is called the tangent developable of γ, that is x (t, u) = γ(t) + u μ (t).

In our previous paper, we have defined pedal curves of framed curves by orthogonal projection.

Definition 2.2

([12]). Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve. p ∈ R 3 is a fixed point. The pedal curve P e γ , p : I → R 3 of (γ, n 1, n 2) relative to p is

P e γ , p ( t ) = p − ( ( p − γ ( t ) ) ⋅ n 2 ( t ) ) n 2 ( t ) .

We call p the pedal point.

Definition 2.3

([27]). U is an open domain in R 2 . ( x , n , s ) : U → R 3 × Δ is called a framed surface if x u (u, v) ⋅ n (u, v) = x v (u, v) ⋅ n (u, v) = 0 for all (u, v) ∈ U, where x u ( u , v ) = ∂ x ∂ u ( u , v ) and x v ( u , v ) = ∂ x ∂ v ( u , v ) .

Let F : J × R 3 → R , ( t , x ) ↦ F ( t , x ) be a smooth map, where J is an interval of R . Then F = 0 means a one-parameter family of surfaces in R 3 . If there exists a surface S tangent to each member of the family at some point and any point on S is on one of surfaces in the family, then S is called the envelope of the family of surfaces [28]. One can get the envelope by eliminating the parameter t from F ( t , x ) = ∂ F ∂ t ( t , x ) = 0 .

3 Pedal and negative pedal surfaces

Definition 3.1.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 .

  1. The pedal surface P S γ , p : I × [ 0,2 π ] → R 3 of (γ, n 1, n 2) relative to p is

    P S γ , p ( t , θ ) = p + ( γ ( t ) − p ) ⋅ n 1 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 cos ⁡ θ 2 n 1 ( t ) + ( γ ( t ) − p ) ⋅ n 2 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 sin ⁡ θ 2 n 2 ( t ) .

  2. Suppose the point p satisfies (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I. The negative pedal surface N P S γ , p : I × R → R 3 of (γ, n 1, n 2) relative to p is

    N P S γ , p ( t , λ ) = 2 γ ( t ) − p + λ n 1 ( t ) − ( γ ( t ) − p ) ⋅ ( γ ( t ) − p ) + λ ( γ ( t ) − p ) ⋅ n 1 ( t ) ( γ ( t ) − p ) ⋅ n 2 ( t ) n 2 ( t ) .

Remark 3.2.

If the point p is on the tangent developable of γ, then the pedal surface P S γ , p is not a smooth surface.

When the curve γ is regular, the two surfaces defined in Definition 3.1 can be both regarded as envelopes of families of surfaces.

Proposition 3.3.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 , where α(t) ≠ 0.

  1. The pedal surface P S γ , p of (γ, n 1, n 2) relative to p is the envelope of a family of spheres.

  2. If the point p satisfies (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I, then the negative pedal surface N P S γ , p of (γ, n 1, n 2) relative to p is the envelope of a family of planes.

Proof.

(a) Define F : I × R 3 → R by F(t, x ) = (γ(t) − x ) ⋅ ( x − p ). It is clear that F = 0 is a family of spheres and

∂ F ∂ t ( t , x ) = α ( t ) μ ( t ) ⋅ ( x − p ) .

Let F = ∂ F ∂ t = 0 , then there exists θ ∈ [0, 2π] such that

x − p = ( γ ( t ) − p ) ⋅ n 1 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 cos ⁡ θ 2 n 1 ( t ) + ( γ ( t ) − p ) ⋅ n 2 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 sin ⁡ θ 2 n 2 ( t ) .

  1. Define G : I × R 3 → R by G(t, x ) = (γ(t) − x ) ⋅ (γ(t) − p ). It is clear that G = 0 is a family of planes and

∂ G ∂ t ( t , x ) = α ( t ) μ ( t ) ⋅ ( 2 γ ( t ) − p − x ) .

Let G = ∂ G ∂ t = 0 , then there exists λ ∈ R such that

x = 2 γ ( t ) − p + λ n 1 ( t ) − ( γ ( t ) − p ) ⋅ ( γ ( t ) − p ) + λ ( γ ( t ) − p ) ⋅ n 1 ( t ) ( γ ( t ) − p ) ⋅ n 2 ( t ) n 2 ( t ) .

□

Definition 3.4.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 satisfy (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I. The negative pedal curve of (γ, n 1, n 2) relative to p is

N P e γ , p ( t ) = 2 γ ( t ) − p − 2 α ( t ) M ( t ) n 1 ( t ) − 1 ( γ ( t ) − p ) ⋅ n 2 ( t ) ( γ ( t ) − p ) ⋅ ( γ ( t ) − p ) − 2 α ( t ) M ( t ) ( γ ( t ) − p ) ⋅ n 1 ( t ) n 2 ( t ) .

Now we consider singularities of the pedal surface and negative pedal surface.

Theorem 3.5.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 .

  1. Suppose that the point p is not on the tangent developable of γ. P S γ , p is the pedal surface of (γ, n 1, n 2) relative to p . The singular locus of P S γ , p coincides with the pedal curve P e γ , p and the point p .

  2. Suppose that the point p satisfies (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I. N P S γ , p is the negative pedal surface of (γ, n 1, n 2) relative to p . The singular locus of N P S γ , p coincides with the negative pedal curve N P e γ , p .

Proof.

(a) By calculation, we have

(1) 2 ∂ P S γ , p ∂ t ( t , θ ) = M ( t ) ( γ ( t ) − p ) ⋅ n 1 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 cos ⁡ θ μ ( t ) + M ( t ) ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) ( ( γ ( t ) − p ) ⋅ μ ( t ) ) ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( n 1 ( t ) cos ⁡ θ + n 2 ( t ) sin ⁡ θ )   + L ( t ) ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( − n 1 ( t ) sin ⁡ θ + n 2 ( t ) cos ⁡ θ ) + M ( t ) ( ( γ ( t ) − p ) ⋅ μ ( t ) ) n 1 ( t ) ,

(2) ∂ P S γ , p ∂ θ ( t , θ ) = 1 2 ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( − n 1 ( t ) sin ⁡ θ + n 2 ( t ) cos ⁡ θ )

and

∂ P S γ , p ∂ t ( t , θ ) × ∂ P S γ , p ∂ θ ( t , θ ) = 1 4 M ( t ) ( γ ( t ) − p ) ⋅ n 1 ( t ) + ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 cos θ   − ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( n 1 ( t ) cos θ + n 2 ( t ) sin θ )   + ( ( γ ( t ) − p ) ⋅ μ ( t ) ) μ ( t ) .

So (t 0, θ 0) is a singularity of P S γ , p if and only if

( γ ( t 0 ) − p ) ⋅ n 1 ( t 0 ) + ( ( γ ( t 0 ) − p ) ⋅ n 1 ( t 0 ) ) 2 + ( ( γ ( t 0 ) − p ) ⋅ n 2 ( t 0 ) ) 2 cos θ 0 = 0 .

This means that

P S γ , p ( t 0 , θ 0 ) = p + ( ( γ ( t 0 ) − p ) ⋅ n 2 ( t 0 ) ) n 2 ( t 0 )

or

P S γ , p ( t 0 , θ 0 ) = p .

  1. By calculation, we have

(3) ∂ N P S γ , p ∂ t ( t , λ ) = L ( t ) ( γ ( t ) − p ) ⋅ ( γ ( t ) − p ) + λ ( γ ( t ) − p ) ⋅ n 1 ( t ) ( γ ( t ) − p ) ⋅ n 2 ( t ) n 1 ( t ) + ( 2 α ( t ) + λ M ( t ) ) μ ( t )   − 1 ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( 2 α ( t ) + λ M ( t ) ) ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) ( ( γ ( t ) − p ) ⋅ μ ( t ) )   + L ( t ) ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) ( ( γ ( t ) − p ) ⋅ ( γ ( t ) − p ) + λ ( γ ( t ) − p ) ⋅ n 1 ( t ) ) n 2 ( t ) ,

(4) ∂ N P S γ , p ∂ λ ( t , λ ) = n 1 ( t ) − ( γ ( t ) − p ) ⋅ n 1 ( t ) ( γ ( t ) − p ) ⋅ n 2 ( t ) n 2 ( t )

and

∂ N P S γ , p ∂ t ( t , λ ) × ∂ N P S γ , p ∂ λ ( t , λ ) = 2 α ( t ) + λ M ( t ) ( γ ( t ) − p ) ⋅ n 2 ( t ) ( γ ( t ) − p ) .

So (t 0, λ 0) is a singularity of N P S γ , p if and only if 2α(t 0) + λ 0 M(t 0) = 0. This means that

N P S γ , p ( t 0 , λ 0 ) = 2 γ ( t 0 ) − p − 2 α ( t 0 ) M ( t 0 ) n 1 ( t 0 ) − ( γ ( t 0 ) − p ) ⋅ ( γ ( t 0 ) − p ) − 2 α ( t 0 ) M ( t 0 ) ( γ ( t 0 ) − p ) ⋅ n 1 ( t 0 ) ( γ ( t 0 ) − p ) ⋅ n 2 ( t 0 ) n 2 ( t 0 ) .

□

The pedal surface and negative pedal surface may have singular points. We give conditions when they are framed base surfaces. For framed base surfaces, please see [27].

Proposition 3.6.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 .

  1. If p is not on the tangent developable of γ, then the pedal surface P S γ , p is a framed base surface.

  2. Suppose that the point p satisfies (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I. Then the negative pedal surface N P S γ , p is a framed base surface.

Proof.

(a) Take

n ( t , θ ) = 1 ‖ γ ( t ) − p ‖ ( ( γ ( t ) − p ) ⋅ μ ( t ) ) μ ( t )   − ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 ( n 1 ( t ) cos ⁡ θ + n 2 ( t ) sin ⁡ θ )

and

s ( t , θ ) = n 1 ( t ) sin ⁡ θ − n 2 ( t ) cos ⁡ θ .

By Equations (1) and (2), we can check that

∂ P S γ , p ∂ t ( t , θ ) ⋅ n ( t , θ ) = ∂ P S γ , p ∂ θ ( t , θ ) ⋅ n ( t , θ ) = 0 .

So ( P S γ , p , n , s ) is a framed surface.

  1. Take

N ( t , λ ) = γ ( t ) − p ‖ γ ( t ) − p ‖

and

S ( t , λ ) = ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) n 1 ( t ) − ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) n 2 ( t ) ( ( γ ( t ) − p ) ⋅ n 1 ( t ) ) 2 + ( ( γ ( t ) − p ) ⋅ n 2 ( t ) ) 2 .

By Equations (3) and (4), we can check that

∂ N P S γ , p ∂ t ( t , λ ) ⋅ N ( t , λ ) = ∂ N P S γ , p ∂ λ ( t , λ ) ⋅ N ( t , λ ) = 0 .

So ( N P S γ , p , N , S ) is a framed surface. □

Following propositions claim a fact that the negative pedal curve is also a framed base curve and the relation between the negative pedal curve and the pedal curve.

Proposition 3.7.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve with the curvature (L, M, 0, α) and p ∈ R 3 satisfy (γ(t) − p ) ⋅ n 2(t) > 0 for all t ∈ I. N P e γ , p is the negative pedal curve of (γ, n 1, n 2) relative to p . Let

n ̄ 1 ( t ) = ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n 2 ) 2 ‖ γ − p ‖ μ − ( γ − p ) ⋅ μ ‖ γ − p ‖ ( ( γ − p ) ⋅ n 1 ) n 1 + ( ( γ − p ) ⋅ n 2 ) n 2 ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n 2 ) 2 ( t ) , n ̄ 2 ( t ) = γ − p ‖ γ − p ‖ ( t ) .

Then ( N P e γ , p , n ̄ 1 , n ̄ 2 ) is a framed curve with the curvature ( L ̄ , M ̄ , 0 , α ̄ ) , where

L ̄ ( t ) = − α ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n 2 ) 2 ( γ − p ) ⋅ ( γ − p ) ( t ) , M ̄ ( t ) = M ‖ γ − p ‖ ( γ − p ) ⋅ n 2 ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n 2 ) 2 ( t ) , α ̄ ( t ) = g ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n 2 ) 2 ( t ) , g ( t ) = − 2 α ′ M − 2 α M ′ M 2 ( γ − p ) ⋅ n 2 + L ( γ − p ) ⋅ ( γ − p ) ( ( γ − p ) ⋅ n 2 ) 2 − 2 α L ( γ − p ) ⋅ n 1 M ( ( γ − p ) ⋅ n 2 ) 2 ( t ) .

Proof.

We have

N P e γ , p ′ ( t ) = g ( ( γ − p ) ⋅ n 2 ) n 1 − ( ( γ − p ) ⋅ n 1 ) n 2 ( t ) .

It is obvious that N P e γ , p ′ ( t ) ⋅ n ̄ 1 ( t ) = N P e γ , p ′ ( t ) ⋅ n ̄ 2 ( t ) = 0 . So ( N P e γ , p , n ̄ 1 , n ̄ 2 ) is a framed curve. Let

μ ̄ ( t ) = n ̄ 1 ( t ) × n ̄ 2 ( t ) = − ( ( γ − p ) ⋅ n 2 ) n 1 + ( ( γ − p ) ⋅ n 1 ) n 2 ( ( γ − p ) ⋅ n 1 ) 2 + ( ( γ − p ) ⋅ n ̄ 2 ) 2 ( t ) .

Note that

L ̄ ( t ) = n ̄ 1 ′ ( t ) ⋅ n ̄ 2 ( t ) , M ̄ ( t ) = n ̄ 1 ′ ( t ) ⋅ μ ̄ ( t ) , α ̄ ( t ) = N P e γ , p ′ ( t ) ⋅ μ ̄ ( t )

and

n ̄ 2 ′ ( t ) ⋅ μ ̄ ( t ) = 0 .

We complete the proof. □

Proposition 3.8.

Let ( γ , n 1 , n 2 ) : I → R 3 × Δ be a framed curve and p ∈ R 3 satisfy (γ(t) − p ) ⋅ n 2(t) ≠ 0 for all t ∈ I. N P e γ , p is the negative pedal curve of (γ, n 1, n 2) relative to p . Then the pedal curve of N P e γ , p relative to p coincides with γ.

Proof.

By Proposition 3.7, we know the binormal direction of N P e γ , p is

n ̄ 2 ( t ) = sign ( γ ( t ) − p ) ⋅ n 2 ( t ) γ ( t ) − p ‖ γ ( t ) − p ‖ ,

where sign is the signature function. Note that

( N P e γ , p ( t ) − p ) ⋅ n ̄ 2 ( t ) = 2 γ − 2 p − 2 α M n 1 − ( γ − p ) ⋅ ( γ − p ) − 2 α M ( γ − p ) ⋅ n 1 ( γ − p ) ⋅ n 2 n 2 ⋅ n ̄ 2 ( t ) = sign ( γ ( t ) − p ) ⋅ n 2 ‖ γ ( t ) − p ‖ .

So the pedal curve of N P e γ , p relative to p is

P e N P e γ , p , p ( t ) = p − ( ( p − N P e γ , p ( t ) ) ⋅ n ̄ 2 ( t ) ) n ̄ 2 ( t ) = p + ‖ γ ( t ) − p ‖ γ ( t ) − p ‖ γ ( t ) − p ‖ = γ ( t ) .

□

4 Examples

Here, we give two examples to show pedal and negative pedal surfaces of framed curves.

Example 4.1.

Let ( γ , n 1 , n 2 ) : ( 0 , + ∞ ) → R 3 × Δ be

γ ( t ) = ( cos ⁡ t , sin ⁡ t , t ) , n 1 ( t ) = ( cos ⁡ t , sin ⁡ t , 0 ) , n 2 ( t ) = 2 2 ( − sin ⁡ t , cos ⁡ t , − 1 )

with the curvature L ( t ) = 2 2 , M ( t ) = 2 2 , α ( t ) = 2 .

Let p = 0. The pedal curve, negative pedal curve, pedal surface and negative pedal surface of (γ, n 1, n 2) relative to p are

P e γ , p ( t ) = − t 2 ( − sin ⁡ t , cos ⁡ t , − 1 ) , N P e γ , p ( t ) = ( − 2 ⁡ cos ⁡ t , − 2 ⁡ sin ⁡ t , 2 t ) + t 2 − 3 t ( − sin ⁡ t , cos ⁡ t , − 1 ) , P S γ , p ( t , θ ) = 2 4 2 + 2 + t 2 cos ⁡ θ ( cos ⁡ t , sin ⁡ t , 0 ) + − t + 2 + t 2 sin ⁡ θ 4 ( − sin ⁡ t , cos ⁡ t , − 1 )

and

N P S γ , p ( t , λ ) = ( 2 ⁡ cos ⁡ t , 2 ⁡ sin ⁡ t , 2 t ) + λ ( cos ⁡ t , sin ⁡ t , 0 ) + t 2 + 1 + λ t ( − sin ⁡ t , cos ⁡ t , − 1 ) ,

respectively (Figures 2 and 3).

Figure 2: γ (black), its pedal curve (red) and pedal surface (mesh).
Figure 2:

γ (black), its pedal curve (red) and pedal surface (mesh).

Figure 3: γ (black), its negative pedal curve (red) and negative pedal surface (mesh).
Figure 3:

γ (black), its negative pedal curve (red) and negative pedal surface (mesh).

Example 4.2.

Let ( γ , n 1 , n 2 ) : − 2 , 2 → R 3 × Δ be

γ ( t ) = 3 ( t ⁡ sin ⁡ t + cos ⁡ t ) , 3 ( t ⁡ cos ⁡ t − sin ⁡ t ) , 2 t 2 , n 1 ( t ) = ( − sin ⁡ t , − cos ⁡ t , 0 ) , n 2 ( t ) = 1 5 ( − 4 ⁡ cos ⁡ t , 4 ⁡ sin ⁡ t , 3 )

with the curvature L ( t ) = 4 5 , M ( t ) = 3 5 , α ( t ) = − 5 t .

Let p = 0. The pedal curve, negative pedal curve, pedal surface and negative pedal surface of (γ, n 1, n 2) relative to p are

P e γ , p ( t ) = 6 t 2 − 12 25 ( − 4 ⁡ cos ⁡ t , 4 ⁡ sin ⁡ t , 3 ) , N P e γ , p ( t ) = − 32 3 t ⁡ sin ⁡ t + 6 ⁡ cos ⁡ t , − 32 3 t ⁡ cos ⁡ t − 6 ⁡ sin ⁡ t , 4 t 2 + 4 t 4 − 41 t 2 + 9 6 t 2 − 12 ( − 4 ⁡ cos ⁡ t , 4 ⁡ sin ⁡ t , 3 ) , P S γ , p ( t , θ ) = − 15 t + 36 t 4 + 81 t 2 + 144 cos ⁡ θ 10 ( − sin ⁡ t , − cos ⁡ t , 0 ) + 6 t 2 − 12 + 36 t 4 + 81 t 2 + 144 sin ⁡ θ 50 ( − 4 ⁡ cos ⁡ t , 4 ⁡ sin ⁡ t , 3 )

and

N P S γ , p ( t , λ ) = 6 ( t ⁡ sin ⁡ t + cos ⁡ t ) , 6 ( t ⁡ cos ⁡ t − sin ⁡ t ) , 4 t 2 − λ ( sin ⁡ t , cos ⁡ t , 0 ) − 4 t 4 + 9 t 2 + 9 − 3 t λ 6 t 2 − 12 ( − 4 ⁡ cos ⁡ t , 4 ⁡ sin ⁡ t , 3 ) ,

respectively (Figures 4 and 5).

Figure 4: γ (black), its pedal curve (red) and pedal surface (mesh).
Figure 4:

γ (black), its pedal curve (red) and pedal surface (mesh).

Figure 5: γ (black), its negative pedal curve (red) and negative pedal surface (mesh).
Figure 5:

γ (black), its negative pedal curve (red) and negative pedal surface (mesh).

From above examples, we can see the singular locus of the pedal surface and the negative pedal surface. This is consistent with Theorem 3.5.

5 Conclusions

We define pedal and negative pedal surfaces of framed curves in the Euclidean 3-space. Their singular loci are pedal and negative pedal curves, respectively. The two surfaces are envelopes of families of spheres and planes, respectively. Moreover, we also give sufficient conditions when they can be framed base surfaces. As for negative pedal curves, we give the sufficient condition when they can be framed base curves and discuss their connection with pedal curves. The theoretical value of our research is the correspondence between curves and surfaces, which is different from the correspondence between manifolds with the same dimension in most previous studies.

Corresponding author: Donghe Pei, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R. China, E-mail: 

Acknowledgments

The authors would like to thank the reviewers for helpful comments to improve the original manuscript.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: Authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. KY wrote the main manuscript text. KY and DP reviewed and edited. DP funded.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: Authors declare that there is no conflict of interest.

  6. Research funding: This work was supported by the National Natural Science Foundation of China (Grant No. 11671070).

  7. Data availability: Not applicable.

References

[1] S. Honda and M. Takahashi, Evolutes and focal surfaces of framed immersions in the Euclidean space, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 1, 497–516, https://doi.org/10.1017/prm.2018.84.Suche in Google Scholar

[2] T. Fukunaga and M. Takahashi, Existence and uniqueness for Legendre curves, J. Geom. 104 (2013), no. 2, 297–307, https://doi.org/10.1007/s00022-013-0162-6.Suche in Google Scholar

[3] S. Honda and M. Takahashi, Framed curves in the Euclidean space, Adv. Geom. 16 (2016), no. 3, 265–276, https://doi.org/10.1515/advgeom-2015-0035.Suche in Google Scholar

[4] R. L. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly 82 (1975), 246–251, https://doi.org/10.2307/2319846.Suche in Google Scholar

[5] S. Honda and M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish J. Math. 44 (2020), no. 3, 883–899, https://doi.org/10.3906/mat-1905-63.Suche in Google Scholar

[6] F. Kuczmarski, Roads and wheels, roulettes and pedals, Amer. Math. Monthly 118 (2011), no. 6, 479–496, https://doi.org/10.4169/amer.math.monthly.118.06.479.Suche in Google Scholar

[7] J. Stávek, Kepler’s ellipse observed from Newton’s evolute (1687), Horrebow’s circle (1717), Hamilton’s pedal curve (1847), and two contrapedal curves (28.10.2018), Appl. Phys. Res. 10 (2018), no. 6, 90–101, https://doi.org/10.5539/apr.v10n6p90.Suche in Google Scholar

[8] T. Nishimura, Normal forms for singularities of pedal curves produced by non-singular dual curve germs in Sn, Geom. Dedicata 133 (2008), 59–66, https://doi.org/10.1007/s10711-008-9233-5.Suche in Google Scholar

[9] T. Nishimura, Singularities of pedal curves produced by singular dual curve germs in Sn, Demonstr. Math. 43 (2010), no. 2, 447–459, https://doi.org/10.1515/dema-2010-0216.Suche in Google Scholar

[10] Y. Li and D. Pei, Pedal curves of frontals in the Euclidean plane, Math. Methods Appl. Sci. 41 (2018), no. 5, 1988–1997, https://doi.org/10.1002/mma.4724.Suche in Google Scholar

[11] O. O. Tuncer, H. Ceyhan, İ. Gök, and F. N. Ekmekci, Notes on pedal and contrapedal curves of fronts in the Euclidean plane, Math. Methods Appl. Sci. 41 (2018), no. 13, 5096–5111, https://doi.org/10.1002/mma.5056.Suche in Google Scholar

[12] K. Yao, M. Li, E. Li, and D. Pei, Pedal and contrapedal curves of framed immersions in the Euclidean 3-space, Mediterr. J. Math. 20 (2023), no. 4, 204, https://doi.org/10.1007/s00009-023-02408-z.Suche in Google Scholar

[13] E. Li and D. Pei, Involute-evolute and pedal-contrapedal curve pairs on S2, Math. Methods Appl. Sci. 45 (2022), no. 18, 11986–12000, https://doi.org/10.1002/mma.6994.Suche in Google Scholar

[14] G. Aydın Şekerci, On evolutoids and pedaloids in Minkowski 3-space, J. Geom. Phys. 168 (2021), 104313, https://doi.org/10.1016/j.geomphys.2021.104313.Suche in Google Scholar

[15] Y. Li and O. O. Tuncer, On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space, Math. Methods Appl. Sci. 46 (2023), no. 9, 11157–11171, https://doi.org/10.1002/mma.9173.Suche in Google Scholar

[16] Y. Li, Y. Zhu, and Q. Sun, Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space, Int. J. Geom. Methods Mod. Phys. 18 (2021), no. 1, 2150008. https://doi.org/10.1142/S0219887821500080.Suche in Google Scholar

[17] G. Aydın Şekerci and S. Izumiya, Evolutoids and pedaloids of Minkowski plane curves, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 5, 2813–2834, https://doi.org/10.1007/s40840-021-01091-1.Suche in Google Scholar

[18] S. Izumiya and N. Takeuchi, Evolutoids and pedaloids of plane curves, Note Mat. 39 (2019), no. 2, 13–23, https://doi.org/10.1285/i15900932v39n2p13.Suche in Google Scholar

[19] Z. Yang, Y. Li, M. Erdoğdu, and Y. Zhu, Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane, J. Geom. Phys. 176 (2022), 104513, https://doi.org/10.1016/j.geomphys.2022.104513.Suche in Google Scholar

[20] P. Blaschke, Pedal coordinates, dark Kepler, and other force problems, J. Math. Phys. 58 (2017), no. 6, 063505, https://doi.org/10.1063/1.4984905.Suche in Google Scholar

[21] P. Blaschke, Pedal coordinates, solar sail orbits, Dipole drive and other force problems, J. Math. Anal. Appl. 506 (2022), no. 1, 125537, https://doi.org/10.1016/j.jmaa.2021.125537.Suche in Google Scholar

[22] P. Blaschke, F. Blaschke, and M. Blaschke, Pedal coordinates and free double linkage, J. Geom. Phys. 171 (2022), 104397, https://doi.org/10.1016/j.geomphys.2021.104397.Suche in Google Scholar

[23] V. I. Arnol’d, Dynamical Systems. VIII, Springer-Verlag, Berlin, 1993.Suche in Google Scholar

[24] S. Izumiya and N. Takeuchi, Primitivoids and inversions of plane curves, Beitr. Algebra Geom. 61 (2020), no. 2, 317–334, https://doi.org/10.1007/s13366-019-00472-9.Suche in Google Scholar

[25] S. Janeczko and T. Nishimura, Anti-orthotomics of frontals and their applications, J. Math. Anal. Appl. 487 (2020), no. 2, 124019, https://doi.org/10.1016/j.jmaa.2020.124019.Suche in Google Scholar

[26] M. Abdelatif, H. Nour Alldeen, H. Saoud, and S. Suorya, Finite type of the pedal of revolution surfaces in E3, J. Korean Math. Soc. 53 (2016), no. 4, 909–928. https://doi.org/10.4134/JKMS.j150336.Suche in Google Scholar

[27] T. Fukunaga and M. Takahashi, Framed surfaces in the Euclidean space, Bull. Braz. Math. Soc. (N.S.) 50 (2019), no. 1, 37–65, https://doi.org/10.1007/s00574-018-0090-z.Suche in Google Scholar

[28] A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Chapman & Hall/CRC, Boca Raton, FL, 2006.Suche in Google Scholar
Received: 2025-01-10
Accepted: 2025-09-18
Published Online: 2025-11-13

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0206
Schlagwörter für diesen Artikel
pedal surface; negative pedal surface; negative pedal curve; framed curve; framed surface
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Artikel in diesem Heft

  1. On I-convergence of nets of functions in fuzzy metric spaces
  2. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  3. Forbidden subgraphs of TI-power graphs of finite groups
  4. Finite group with some c#-normal and S-quasinormally embedded subgroups
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  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Simplicial complexes defined on groups
  8. Further results on permanents of Laplacian matrices of trees
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  10. Classes of modules closed under projective covers
  11. On the dimension of the algebraic sum of subspaces
  12. Green's graphs of a semigroup
  13. On an uncertainty principle for small index subgroups of finite fields
  14. On a generalization of I-regularity
  15. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  16. The hyperbolic CS decomposition of tensors based on the C-product
  17. On weakly classical 1-absorbing prime submodules
  18. Equational characterizations for some subclasses of domains
  19. Algebraic Geometry
  20. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  21. Embedding of lattices and K3-covers of an Enriques surface
  22. Kodaira-Spencer maps for elliptic orbispheres as isomorphisms of Frobenius algebras
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  25. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
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  27. The higher mapping cone axiom
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  31. Some new bounds on resolvent energy of a graph
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  54. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
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  58. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
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  60. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
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  82. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  83. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  84. q-Stirling sequence spaces associated with q-Bell numbers
  85. Chlodowsky variant of Bernstein-type operators on the domain
  86. Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol
  87. Characterization of derivations on strongly double triangle subspace lattice algebras by local actions
  88. Fixed point approaches to the stability of Jensen’s functional equation
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  111. Recurrence relations satisfied by some arithmetic groups
  112. The hybrid power mean involving the Kloosterman sums and Dedekind sums
  113. Numerical Methods
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  115. A kind of univariate improved Shepard-Euler operators
  116. Probability and Statistics
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  118. Multiple G-Stratonovich integral in G-expectation space
  119. p-variation and Chung's LIL of sub-bifractional Brownian motion and applications
  120. Real Analysis
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  122. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  123. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  124. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  125. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  126. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  127. Topology
  128. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  129. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  130. On SI2-convergence in T0-spaces
  131. Generalized quandle polynomials and their applications to stuquandles, stuck links, and RNA folding
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