Nonparametric Bézier Representation of Polynomial Transition Curves

Sánchez-Reyes, J.; Chacón, J. M.

doi:10.1061/(ASCE)SU.1943-5428.0000251

Open access

Technical Papers

Jan 24, 2018

Nonparametric Bézier Representation of Polynomial Transition Curves

Authors: J. Sánchez-Reyes https://orcid.org/0000-0003-2080-7269 [email protected] and J. M. ChacónAuthor Affiliations

Publication: Journal of Surveying Engineering

Volume 144, Issue 2

https://doi.org/10.1061/(ASCE)SU.1943-5428.0000251

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Abstract

In a series of papers in this journal, polynomial solutions of transition curves are introduced for road design, constructed as graphs of polynomial functions. These curves admit a simple expression in nonparametric Bézier form, which facilitates their implementation into commercial software and data exchange. Furthermore, in this Bézier representation, few compact formulas describe all possible cases, and the boundary conditions defining the curves translate into intuitive geometric arrangements of the control points. This is illustrated with several examples, including cases where, at the endpoints, zero curvature and smoothness of the curvature diagram are required. The authors also consider an alternative using a nonparametric B-spline curve of lower degree, but at the cost of a more complex model. Quartic splines, at a minimum, are required to ensure a continuous change of lateral acceleration along the transition.

Introduction

Transition Curves

Transition curves are used in the route design of highways and railways, or trajectories for automatic guided vehicles, to achieve smooth transitions between lines and/or circles. The book recently published by Kobryń (2017a) provides an updated survey on the topic. The clothoid, known variously as the spiral of Cornu or Euler’s spiral (Lawrence 1972), is a classical transition curve, because it enjoys the remarkable property of a curvature that varies linearly with arc length. However, the clothoid is not the option of choice in terms of comfort, in particular regarding the change of lateral acceleration. Thus, Baykal et al. (1997) and Bosurgi and D’Andrea (2012) advocate a curvature given by quintic polynomial functions. The resulting transition curve is more flexible, displays advantages regarding ride quality, and can be optimized from a dynamic perspective (Bosurgi et al. 2016). Tari and Baykal (2005) explored the additional possibilities resulting from transition curves with a curvature defined by a septic polynomial.

Unfortunately, these curves involve transcendental functions in their definition that do not admit a rational parametrization, and are hence not expressible in the nonuniform rational B-spline (NURBS) standard used in commercial computer aided-design (CAD) programs (Farin 1999). Certainly, this problem could be circumvented by finding a suitable arc spline (Meek and Walton 2004) or polynomial approximation, thus expressible as NURBS. This has been done for the clothoid (Wang et al. 2001) by blending of Taylor expansions, but this solution yields a high polynomial degree. As a more efficient alternative, Sánchez-Reyes and Chacón (2003) advocate a two-point Hermite expansion, which in the quintic case results in control points admitting straightforward explicit expressions (Sánchez-Reyes and Chacón 2015).

An alternative is to abandon the clothoid or any transcendental curve altogether and resort to polynomial or rational parametric curves {x(t),y(t)}, where x(t),y(t) denote polynomial or rational functions, using Hermite interpolation to obtain transitions with curvature continuity. There exists a rich literature on the subject, focusing on good curvature distribution (typically monotone curvature, i.e., spirals). Different options are available, including a polynomial cubic (Habib and Sakai 2009), a rational cubic (Dietz et al. 2008; Habib and Sakai 2010), a pair of Bézier quartics (Ahmad and Ali 2013), or rational quartics obtained by inverting a conic (Kurnosenko 2010). One can even define spiral transitions with rational offsets [Walton and Meek (2013) and references therein], a desirable property in highway design, because the borders are also expressible in rational Bézier form.

Explicit Functions for Defining Transition Curves

Recently, Kobryń (2011, 2014, 2016a, 2017a) proposed a novel approach, namely, using graphs [y = y(x)] of degree-n polynomial functions as transition curves for route design, or for designing vertical arcs in highway profiles (Kobryń 2016b, 2017b). These curves, given in explicit form (Rockwood and Chambers 1996; Patrikalakis and Maekawa 2002), are called nonparametric or functional (Farin 2002). Compared with general parametric curves, this model enjoys definite advantages:

•

Simpler representation and tracing: Only one scalar function [y(x)] is needed, instead of two components. This is the natural choice for vertical transition curves, requiring an explicit function of the height [y(x)] in terms of the horizontal coordinate (x).

•

Simple implicit form: y–y(x) = 0. Thus, given an arbitrary point (p), determining whether p lies on, above, or below the curve is immediate. This point membership classification plays a key role in graphics processing.

•

Straightforward inversion algorithm: If a point p = {x,y} lies on the curve, the corresponding curve parameter coincides with its abscissa (x).

•

More predictable shape: Single-valued curves are free from self-intersections.

•

Absence of singular points: The graph of the polynomial is also free from cusps, as the derivative vector never vanishes. In general, a smooth C^r function [y(x)] always guarantees a graph with the same smoothness [i.e., with geometric continuity (G^r) (Farin 2002)].

However, nonparametric curves offer less flexibility:

•

They are axis-dependent: A particular system of coordinates {x,y} must be chosen (Fig. 1).

•

The tangent angles θ₀,θ_n at the endpoints, from the x-axis, are constrained to the interval θ₀,θ_n ∈ (–90°,90°).

Fig. 1. Geometry of a quintic (n = 5) nonparametric Bézier curve [b(t)]

Bézier Representation of Polynomial Functions

In Kobryń (2011, 2014, 2016a, b, 2017a, b), curves defined by an explicit polynomial function are expressed in the traditional monomial basis (i.e., in power form). However, in computer-aided geometric design (CAGD) and computer graphics, the Bernstein basis (Bézier form) is that of choice for its superior geometric and numerical properties (Farin 2002; Farouki 2012). In particular, it enjoys optimality regarding shape preservation among all polynomial bases (Peña 1999), and imposing continuity conditions at the endpoints of a segment is simpler than in the monomial basis. The main goal of this paper is noting that these transition curves admit a straightforward representation in standard Bézier form. This work includes several numerical examples, selected from Kobryń (2011, 2016a).

Bézier Curves Defined

In Bézier form, a polynomial segment [b(t)] of degree n is written as a linear combination of degree-n Bernstein polynomials [

B_{k}^{n} (t)

] and n + 1 control points ({b_k}ⁿ_k=₀)

b (t) = \sum_{k = 0}^{n} b_{k} B_{k}^{n} (t), B_{k}^{n} (t) = (\begin{matrix} n \\ k \end{matrix}) {(1 - t)}^{n - k} t^{k}, t \in [0, 1]

(1)

The subset of nonparametric Bézier curves is characterized by a simple geometric condition, namely, control points (b_k = {x_k,y_k}) constrained to abscissas (x_k) evenly spaced along the x-axis, whereas the y_k values, called Bézier ordinates, can be freely located. Without loss of generality, henceforth the local system of coordinates {x,y} is chosen such that the initial point b₀ has abscissa x₀ = 0. If Δ = x_n denotes the abscissa of the endpoint b_n, then

b_{k} = {x_{k}, y_{k}}, x_{k} = \frac{k}{n} Δ, k = 0, \dots, n

(2)

This geometry is shown in Fig. 1 for the case of a quintic (n=5). Thus, the resulting Bézier curve [Eq. (1)] coincides with the graph of a degree-n polynomial function [y(x)] in Bernstein basis

y (x) = \sum_{k = 0}^{n} y_{k} B_{k}^{n} (t), t = \frac{x}{Δ} \in [0, 1]

(3)

Boundary Conditions in Nonparametric Bézier Form

A transition curve is defined by conditions at the endpoints b₀, b_n, which may include:

•

Positions (ordinates) y₀, y_n;

•

Signed tangential angles θ₀,θ_n (counterclockwise from the x-axis); and

•

Signed curvatures κ₀ = 1/R₀,κ_n = 1/R_n (positive for a convex graph).

This set of conditions is depicted in Fig. 2 at the initial point b₀, and a similar set holds at the endpoint b_n.

Fig. 2. Conditions at initial point b₀ of a nonparametric Bézier curve [b(t)]

Recall two well-known properties of a Bézier curve at the initial point b₀ (Farin 2002):

•

The tangent line (L) is defined by the control leg b₀b₁.

•

The signed distance (h) from b₂ to L (with h > 0 if b₂ lies above) is

h = κ_{0} \frac{n d^{2}}{n - 1}, d = ‖ b_{0} b_{1} ‖

(4)

From these properties, and using basic trigonometry, one derives the ordinates y₀, y₁, and y₂ in terms of the prescribed data {y₀, θ₀,κ₀}

b_{0} = {0, y_{0}}

(5)

b_{1} = {\frac{Δ}{n}, y_{1}}, y_{1} = y_{0} + \frac{Δ}{n} tan θ_{0}

(6)

b_{2} = {\frac{2}{n} Δ, y_{2}}, y_{2} = y_{0} + \frac{2}{n} Δ \cdot tan θ_{0} + h / cos θ_{0}, h = \frac{κ_{0} Δ^{2}}{n (n - 1) {cos}^{2} θ_{0}}

(7)

By the symmetry of the Bézier representation [Eq. (3)], to compute y_n, y_n_–1, and y_n_–2, simply replace the triple {y₀, θ₀,κ₀} with {y_n,180°–θ_n,κ_n}. This is much simpler than in the monomial basis, where a system of equations must be set.

In particular, if the Bézier segment connects with a vanishing curvature [κ₀ = 0 (R₀ = ∞)] to a straight line L (i.e., with G² continuity), then h = 0 [Eq. (7)]. This constraint admits an elegant geometric interpretation as the alignment of b₀, b₁, and b₂ (solid points in Fig. 3) on this tangent line (L). This property extends to a higher smoothness: if the segment connects with G^r continuity to L, then simply set {b₀,…,b_r}, r ≤ n, evenly spaced on L

y_{k} = y_{0} + \frac{k}{n} Δ \cdot tan θ_{0}, k = 0, \dots, r

(8)

Fig. 3. Connecting a Bézier curve to a straight line, with G² and G³ continuity

Fig. 3 illustrates the case r = 3 (smooth curvature), which implies a vanishing third derivative [y′′′(0) = 0].

Kobryń (2011) also analyzes a connection at the endpoint b_n with smooth curvature (G³ continuity when connecting to a circular arc or line) for a horizontal tangent (θ_n = 0). These conditions imply once again a vanishing third derivative [y′′′(1) = 0]. To obtain the resulting ordinate (y_n_–3), consider that the rth derivative at this point is proportional to an rth iterated forward difference (Farin 2002), thereby furnishing the following linear relationship (for r = 3):

y_{n} - 3 y_{n - 1} + 3 y_{n - 2} - y_{n - 3} = 0

(9)

Because θ_n = 0 means that y_n_–1 = y_n, by computing y_n_–2 using the symmetrical version of y₂ [Eq. (7)], one obtains y_n_–3 in terms of κ_n

y_{n - 3} = y_{n} + \frac{3 κ_{n} Δ^{2}}{(n - 1) n}

(10)

Table 1 lists the different endpoint conditions to fulfill for five typical cases (a–e) taken from Kobryń (2011, 2016a), along with the total number m of conditions. A Bézier function [Eq. (3)] has n + 1 degrees of freedom, namely, its n + 1 ordinates {y_k}ⁿ_k=₀ [Eq. (2)], a number that must coincide with m, so the minimum degree required (Table 2) is n = m – 1.

Table 1. Sets of Endpoint Conditions and Their Total Number (m): Nonsmooth Curvature Cases (a, b, and c) and Smooth Curvature (d and e)

Case	Source	y₀	θ₀	R₀	y₀′′′	y_n	θ_n	R_n	y_n′′′	m
		Initial point (t = 0)				Final point (t = 1)
a	Kobryń (2011)	0	θ₀	∞	—	—	0	R_n	—	5
b	Kobryń (2016a)	0	θ₀	R₀	—	y_n	θ_n	—	—	5
c	Kobryń (2016a)	0	θ₀	R₀	—	y_n	θ_n	R_n	—	6
d	Kobryń (2011)	0	θ₀	∞	0	—	0	R_n	0	7
e	Kobryń (2011)	0	θ₀	∞	0	0	θ_n	∞	0	8

Table 2. Minimum Degrees of Bézier and B-Spline Curves Required for Different Endpoint Conditions

Case	m	Degree (n)	N	n	Internal knots
		Bézier	B-spline
a	5	4	4	3	1
b	5	4	4	3	1
c	6	5	5	3	2
d	7	6	6	4	2
e	8	7	7	4	3

B-Spline Alternative

Instead of a Bézier curve [b(t)], a nonparametric B-spline curve [d(t)] provides a lower-degree alternative to solve the interpolation problems posed in the preceding section. To define these curves, the authors adopt the notation used in the textbook on CAGD by Farin (2002). A B-spline curve, of degree n and parameter t ∈ [0,1], is made up of several polynomial segments of the same degree (n), joining at K + 1 knots defined by a nondecreasing knot vector (t = {t_0,…,t_K}). This knot vector is assumed nonperiodic

t = {0 = t_{0} = \dots = t_{n -}_{1}, \dots, t_{K - n}_{+ 1} = \dots = t_{K} = 1}

(11)

with endknots of multiplicity n, so that the curve behaves much like a Bézier curve at the endpoints, being tangent to the control polygon.

Now, the curve has N + 1 control points ({d_k}^N_k=₀) (the so-called de Boor points), where N = K – n + 1. Thus, an additional point for each internal knot is at the designer’s disposal, without increasing the degree. Fig. 4 shows the case of a cubic spline with two internal knots, which has the same number of control points as the quintic Bézier curve of Fig. 1. As in the Bézier case, a local system of coordinates {x,y} is chosen such that the endpoints, now d₀ and d_N, have abscissas ξ₀ = 0 and ξ_N = Δ, respectively. To ensure a nonparametric character, so that x = Δ · t, the de Boor points (d_k = {ξ_k,d_k}) are no longer equally spaced along the x-axis but now have abscissas (ξ_k) proportional to the arithmetic mean of n consecutive knots

d_{k} = {ξ_{k}, d_{k}}, ξ_{k} = \frac{Δ}{n} (t_{k} + \dots + t_{k + n - 1}), k = 0, \dots, N

(12)

Fig. 4. Cubic nonparametric B-spline curve [d(t)]; nonperiodic knot vector t = {0, 0, 0, 1/3, 2/3, 1, 1, 1}

Similar to nonparametric Bézier curves, the de Boor ordinates (d_k) can be freely located.

A B-spline enjoys C^r continuity at an internal knot of multiplicity n – r. As a consequence, at knots of multiplicity one, at least cubics are required to achieve C² continuity. This is the internal continuity the authors assume as reasonable when imposing G² conditions at the endpoints. Similarly, internal C³ continuity is assumed when imposing a G³ condition at either endpoint, which means at least quartics. For the design scenarios listed in Table 1, Table 2 shows these lower B-spline degrees, as well as the number of internal knots to reach the required degrees of freedom.

Because the control polygon is tangent to the curve at the initial point (d₀), Eqs. (5) and (6), now with abscissae [Eq. (12)] and de Boor ordinates, transform to

d_{0} = {0, y_{0}},

d_{1} = {ξ_{1}, d_{1}}, ξ_{1} = \frac{Δ}{n} t_{n}, d_{1} = y_{0} + ξ_{1} tan θ_{0}

(13)

Computing the ordinate d₂ is a straightforward exercise by inserting twice (Farin 2002) the first internal knot (t_n). Thus, in terms of d₁ and d₂, one obtains the Bézier abscissa of the first Bézier segment, over t ∈ [0,t_n], making up the B-spline function. Then, from d₁ [Eq. (13)] and the curvature for this Bézier segment

d_{2} = {ξ_{2}, d_{2}}, d_{2} = y_{0} + ξ_{2} tan θ_{0} + h / cos θ_{0}, h = t_{n} t_{n + 1} \frac{κ_{0} Δ^{2}}{n (n - 1) {cos}^{2} θ_{0}}

(14)

where h = signed distance from d₂ to the tangent line. As in the Bézier case, a smooth G^r connection (r ≤ n) at d₀ with a straight line implies the alignment of points {d₀,…,d_r}, and the ordinates simplify to

d_{k} = y_{0} + ξ_{k} tan θ_{0}, k = 0, \dots, r

(15)

Regarding formulas for d_N, d_N_–1, and d_N_–2, replace the triple {y₀, θ₀,κ₀} with {y_n,180°–θ_n,κ_n}, and derive the symmetrical versions of Eqs. (13), (14), and (15).

Recall that a Bézier curve is nothing other than a B-spline curve defined by the special nonperiodic vector t = {0 = t₀ = ⋯ = t_n–₁, t_n = ⋯ = t₂_n_–1 = 1} [Eq. (11)], without internal knots. In this case, N = n, the de Boor points transform to Bézier points, and Eqs. (12), (13), (14), and (15) transform to Eqs. (2), (6), (7), and (8).

Examples

Figs. 5, 6, and 7 reproduce Cases b, c, and e listed in Tables 1 and 2, with numerical values taken from Kobryń (2011, 2016a) (Table 3):

Fig. 5. Case b: G² interpolation at the initial point, and of position plus tangent at the final point (Bézier quartic; cubic B-spline with an internal knot)

Fig. 6. Case c: G² Hermite interpolation at both endpoints (Bézier quintic; cubic B-spline with two internal knots)

Fig. 7. Case e: Hermite connection with straight lines and G³ continuity (i.e., smooth curvature) (Bézier septic; B-spline quartic with three internal knots)

Table 3. Numerical Values for Cases b, c, and e (Figs. 5, 6, and 7, Respectively)

Case	Source	Δ (m)	y₀	θ₀ (degrees)	R₀ (m)	y₀′′′	y_n (m)	θ_n (degrees)	R_n (m)	y_n′′′
			Initial point (t = 0)				Final point (t = 1)
b	Kobryń (2016a)	500	0	45	−1,000	—	250	26.56	—	—
c	Kobryń (2016a)	559	0	18.43 = 45 – 26.56	−995	—	0	180	497.5	—
e	Kobryń (2011)	1,030.67	0	25	∞	0	0	150	∞	0

Case b: Interpolation of position tangents at both endpoints, but only of curvature κ₀. Data are from Example 1 [universal transition curve (UTC-1)] in Kobryń (2016a).

Case c: Classical G² Hermite interpolation prescribing positions, tangents, and curvatures. Data are from Example 2 (UTC-2) in Kobryń (2016a).

Case e: G³ transition between two lines (i.e., with smooth curvature). Data are from Example 1 (general transition curve) in Kobryń (2011).

In Cases b and c, to find the control points [Bézier (y_k) or de Boor ordinates (d_k)] (Table 4), use Eqs. (6), (7), and (8) in the Bézier case or Eqs. (13) and (14) for B-splines, and their symmetrical counterparts. In Case e, use the simplified versions [Eq. (8) or Eq. (15)]. For simplicity, in the B-spline solution, the internal knots are evenly spaced.

Table 4. Control Points for Cases b, c, and e (Figs. 5, 6, and 7, Respectively)

Case	Curve	Control points (m) and knot vector
b (Fig. 5) Δ = 500	Bézier n = 4	y_k = Δ· ${0, \frac{1}{4}, 0.382, \frac{3}{8}, \frac{1}{2}}$
	B-spline n = 3	d_k = Δ· ${0, \frac{1}{6}, 0.382, \frac{5}{12}, \frac{1}{2}}$
		ξ_k = Δ· ${0, \frac{1}{6}, \frac{1}{2}, \frac{5}{6}, 1}$
		$t = \{0, 0, 0, \frac{1}{2}, 1, 1, 1\}$
c (Fig. 6) Δ = 559	Bézier n = 5	y_k = Δ·{0, 0.067, 0.1, 0.056, 0, 0}
	B-spline n = 3	d_k = Δ·{0, 0.037, 0.086, 0.041, 0, 0}
		ξ_k = Δ· ${0, \frac{1}{9}, \frac{1}{3}, \frac{2}{3}, \frac{8}{9}, 1}$ ,
		$t = \{0, 0, 0, \frac{1}{3}, \frac{2}{3} 1, 1, 1\}$
E (Fig. 7) Δ = 1,030.67	Bézier n = 7	y_k = Δ· ${0, 0.066, 0.133, 0.199, \frac{\sqrt{3}}{7}, \frac{2 \sqrt{3}}{21}, \frac{\sqrt{3}}{21}, 0}$
	B-spline n = 4	d_k = Δ· ${0, 0.029, 0.087, 0.174, \frac{\sqrt{3}}{8}, \frac{\sqrt{3}}{16}, \frac{\sqrt{3}}{48}, 1}$
		ξ_k = $\frac{Δ}{16}$ {0, 1, 3, 6, 10, 13, 15, 16}
		t = ${0, 0, 0, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1, 1, 1}$

For shape analysis, as recommended by Farin (2016) for planar curves, the examples include curvature combs rather than curvature diagrams, the tool usually available in commercial CAD systems. In the general transition curve of Fig. 7, the angles θ₀ and θ_n are within the range (Kobryń 2011) that guarantees the existence of only one extreme E of the curvature in the septic solution. Thus, the curvature increases from zero at the starting point, reaches its maximum at E, and finally decreases to zero at the endpoint. In all three cases, the shape of the Bézier and B-spline solutions are visually indistinguishable, with similar positions for the inflection point (I) in Cases b and c or the extreme E (Case e). However, whereas the Bézier curves have smooth combs in Cases b and c, a cusp appears at the internal knots for the B-spline solution. Indeed, if a curve enjoys G^r continuity, then its curvature comb displays only G^r^–2 continuity. Therefore, the curvature comb of a G² cubic spline (internal knots of multiplicity one) is only G⁰.

The example also plots the lateral change of acceleration (z) for a motion with constant speed (v) exactly along the planar trajectory defined by the function y(x). In terms of successive derivatives (with respect to x)

z = v^{3} \frac{d κ}{d l}, \frac{d κ}{d l} = \frac{y ‴ (1 + y'^{2}) - 3 y' y ″^{2}}{{(1 + y'^{2})}^{3}}

(16)

where l = arc length. The authors considered v = 33.3 m/s (120 km/h) and then plotted z(t), with t = x/Δ ∈ [0,1], as in the examples in Kobryń (2016a). At a point of G^r continuity, the graph z(t) exhibits only G^r^–3 continuity, so the nonsmooth curvature of Cases b and c in general result in discontinuities of z(t) at the endpoints of the transition. The differences between the Bézier and B-spline solutions are more marked because, at the internal knots, z(t) is discontinuous in the cubic splines (Cases b and c). In addition, in Case b, the peak value of z(t) slightly exceeds the recommended maximum for travel comfort (Baykal et al. 1997). In contrast, the Bézier solution [z(t)] shows no internal discontinuities in all examples, albeit in Case e a septic polynomial is needed instead of a quartic spline.

Conclusions

Polynomial transition curves, put forward by Kobryń in this journal, enjoy definite advantages over general parametric curves. Being graphs of polynomial functions, they have a simpler mathematical representation, which is very convenient for the description of vertical transition curves. They display a more predictable shape with no cusps or self-intersections, admit an immediate inversion algorithm, and avoid the computational problems associated with curves defined by curvature functions. However, these curves are axis-dependent, the range of tangent values at the endpoints is restricted, and a curvature function cannot be set.

These transition curves admit a straightforward expression in Bernstein-Bézier form as nonparametric curves, with a small handful of compact formulas encompassing all different cases, in a much simpler way than in the monomial basis. Conditions at the endpoints translate into geometrically intuitive arrangements of the Bézier points. In particular, to impose zero curvature at either endpoint, the three ending Bézier points should be set on the tangent line; to achieve smoothness of the curvature diagram when connecting to a straight line, an additional Bézier point should simply be aligned. By adhering to the Bézier form, the de facto standard for CAD programs, this work allows direct data exchange and the direct incorporation of these transition curves into existing commercial software. It is concluded that this standard form deserves attention for representing polynomial transition curves in surveying engineering.

As a lower-degree alternative to the Bézier model, the authors explore nonparametric B-splines, also a CAD standard. Similarly, the control points derive easily from the endpoint conditions. Additional degrees of freedom are achieved by adding internal knots, without increasing the polynomial degree, at the cost of a piecewise and, hence, more complex model. At a minimum, quartic splines must be used to avoid internal discontinuities in the lateral change of acceleration, and thus to ensure travel comfort.

Notation

The following symbols are used in this paper:

$B_{k}^{n} (t)$: kth Bernstein polynomial of degree n;
b_k: Bézier point;
b(t): Bézier curve;
C^r: parametric continuity of order r;
d_k: de Boor point;
d_k: de Boor ordinate;
d(t): B-spline curve;
E: extremum of maximum curvature;
G^r: geometric continuity of order r;
h: signed distance from b₂ or d₂ to the initial tangent line;
I: inflection point;
K: subscript of last knot;
L: tangent line at b₀;
m: number of conditions;
N: subscript of last de Boor point;
n: degree of a polynomial curve;
p: arbitrary point;
R₀, R_n: signed radii of curvature at endpoints;
t: knot vector;
t: unit parameter, t ∈ [0,1];
t_i: knot;
v: speed along the trajectory;
x_k: Bézier abscissa;
x,y: local coordinates;
y_k: Bézier ordinate;
Δ: abscissa of endpoint b_n or d_N.
θ₀, θ_n: signed angles of the curve end tangents;
κ₀, κ_n: signed curvatures at curve endpoints; and
ξ_k: abscissa of de Boor point d_k.

Acknowledgments

This research was supported by the Spanish Ministerio de Economía y Competitividad (Plan Nacional de I + D + i), under Research Grant DPI2015-65472-R, cofinanced by the European Regional Development Fund (ERDF). The authors are grateful to a reviewer for suggestions, which improved the quality of this article.

References

Ahmad, A., and Ali, J. M. (2013). “Smooth transition curve by planar Bézier quartic.” Sains Malaysiana, 42(7), 989–997.

Abstract

Introduction

Transition Curves

Explicit Functions for Defining Transition Curves

Bézier Representation of Polynomial Functions

Bézier Curves Defined

Boundary Conditions in Nonparametric Bézier Form

B-Spline Alternative

Examples

Conclusions

Notation

Acknowledgments

References

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