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Parallel Helices in Three-Dimensional Space Parallel Helices in Three-Dimensional Space Veronika Chrastinov´a Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics e-mail: [email protected] 1. Parallel curves in the plane. Let p(t) = (x(t), y(t)) be a smooth curve in the plane. Two parallel curves P+(t) = p(t) + rn(t), P−(t) = p(t) − rn(t) (1) at the distance r may be introduced where n(t) is the unit normal vector. They can be alternatively obtained as follows. Let us consider the envelope of circles with center p(t) and radius r. It consists of the points P satisfying d dp(t) (p(t) − P)2 = r2, (p(t) − P)2 = 2 (p(t) − P) = 0. dt dt The second equation implies P − p(t) = cn(t) and then the first one exactly gives formulae (1): the envelope consists of two parallel curves (1). 2. Parallel curves in the space. Let p(t) = (x(t), y(t), z(t)) be a smooth curve. We introduce three equations 2 dp(t) d p(t) dp(t) 2 (p(t) − P)2 = r2, (p(t) − P) = 0, (p(t) − P) + = 0 (2) dt dt2 dt for the point P = P(t). If (31) is regarded as a spherical wave (with center p(t) and radius r), then (32) represents the enveloping surface (the intersection of two infinitesimaly close waves) and (33) the curve of foci P = P(t) (the intersection of three close waves). The final result is independent of the choice of the parameter t. If the arclength t = s is employed, then the Frenet formulae p˙ = t, t˙ = κn, n˙ = −κt + τb, b˙ = −τn ( ˙ = d/ds) easily provide two solutions 1 P (s) = p + kn ± lb (k = , k2 + l2 = r2). (3) ± κ Curves P+(s), P−(s) may be regarded as parallel curves to the curve p(s) in the three- dimensional space. The curves are real if k2 ≤ r2 and imaginary conjugate if k2 > r2. 3. A nontrivial result. The curve P+ (equivalently P−) conversely determines the primary curve p. In more detail: the curve p is conversely parallel to the curve P+ (or: P−) at the distance r. We refer to [1] for a tricky proof and to [2] for many generalizations. At this place we state an instructive direct verification. Given a curve p(s) and the parallel curve P(s) = P+(s), let us calculate two parallel curves P± to the curve P(s). Our aim is to prove that either P+ or P− coincides with the primary curve p = p(s). We wish to determine solution P = P(s) of the system (P(s) − P)2 = r2, P˙ (s)(P(s) − P) = 0, P¨ (s)(P(s) − P) + (P˙ (s))2 = 0 (4) analogous to (2). Using the Frenet formulae and (3), it follows that P˙ = (k˙ − τl)n + (l˙ + τk)b, P¨ = −(k˙ − τl)κt + ((k˙ − τl)˙ − (l˙ + τk)τ)n + ((l˙ + τk)˙ + (k˙ + τl)τ)b. Assuming moreover P = P + at + bn + cb, substitution into (4) gives the system a2 + b2 + c2 = r2, (k˙ − τl)b + (l˙ + τk)c = 0, (k˙ − τl)κa − ((k˙ − τl)· − (l˙ + τk)τ)b − ((l˙ + τk)˙ + (k˙ − τl)τ)c (5) +(k˙ − τl)2 + (l˙ + τk)2 = 0. One can then directly verify that a = 0, b = −k, c = −l is a solution. This provides the curve P = p and the proof is done. Another solution of (5) can be obtained by the substitution b = uk, c = ul. Then (5) turns into the system of two equations a 2 + u2 = 1, (k˙ − τl)κa + ((k˙ − τl)2 + (l˙ + τk)2)(1 + u) = 0 r for the unknown functions u and a, the intersection of an ellipse with a straight line. There are two intersection points. We already know the point a = 0, u = −1, the remaining solu- tion can be easily found and may be omitted here. 4. Parallel helices. Assume κ, τ constant, hence p is a helix. Together with the arclength s along the curve p, we introduce the arclength S along the parallel curve P (P = P+ or P−). Employing (3), the Frenet frame 0 1 P = T = (−ln + kb)(0= d/dS, dS/ds = τr) r 0 1 1 T = (κlt − τkn − τlb) = KN,K = ± pr2(κ2 + τ 2) − 1, τr2 τr2 1 B = β(αt + kN + lb),T = ± τr2 can be found where τr2 rr2(κ2 + τ 2) − 1 r α = , β = ±r = ± pl2 − τ 2/κ2. κl r2κ2 − 1 l We have expressed the curvature and torsion K, T in terms of the primary curvature and torsion κ, τ and they are determined up to a sign ±. If the construction is repeated, then the helices P± parallel to P± again have the original curvature and torsion κ, τ and this can be directly verified. References [1] Chrastinov´a, Parallel curves in three-dimensional space (Sborn´ık5. Konference o matematice a fyzice 2007, UNOB) [2] Chrastinov´a,V., Tryhuk, V., Generalised contact transformations (to appear)..
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