Mullhaupt, Ph.

Infoscience - École polytechnique fédérale de Lausanne

Previous Up Next ArticleCitationsFrom References: 0From Reviews: 0MR2323668 (Review) 53A17Yu¨ce, Salim; Kuruog˘lu, NuriHolditch-type theorems under the closed planar homothetic motions. (English summary)Ital. J. Pure Appl. Math. No. 21 (2007), 105–108.Back in 1858, H. Holditch stated and proved an interesting theorem. An oval path is given togetherwith a rigid rod of length l. Each end of the rod (say A and B) touches the oval. Whenever the rodcompletes a full turn while moving along the oval path, a given fixed point in the rod (say at thedistance a from A and b from B so that l = a+ b) also describes a closed curve, but one which isnot necessarily convex. The surface part between the center of the oval and the curve traced out bythe fixed point on the rod at distance a from A is then removed from the surface bounded by theoval curve. This yields Holditch’s ring, the surface of which obeys a simple and elegant formulaF = abpi.In the paper under review, Holditch-type theorems are presented for closed homothetic motions.For these motions, the linked frame that is attached to the selected moving point in-between Aand B distorts according to a homothetic scale h(t) (i.e. a sort of continuous “zooming in andout” motion). Therefore, two moving frames in the Euclidean planes are defined, one for thefixed reference frame E′ = {O′; e′1, e′2} and one for the moving referential E = {O; e1, e2}. For aclosed homothetic motion, the moving frame is related to the fixed one throughe1 = cos(ϕ(t))e′1 +sin(ϕ(t))e′2,e2 =− sin(ϕ(t))e′1 +cos(ϕ(t))e′2,and the point x′ = (x′1, x′2) ∈ E′ is related to x= (x1, x2) ∈ E throughx′ = h(t)x−(u1(t)u2(t)),where h(t) is the homothetic periodic scale, ϕ(t) is a periodic rotation and u1(t) and u2(t) areperiodic translation functions, that is, there exists T > 0 such that h(t+ T ) = h(t), ϕ(t+ T ) =ϕ(t) + 2ϕν, u1(t+ T ) = u1(t) and u2(t+ T ) = u2(t). The integer ν ∈ Z is called the rotationnumber of the closed planar homothetic motion.The first theorem states that if the points A= (0, 0) and B = (a, 0) ∈ E draw closed-trajectorycurves kA and kB with the orbit areas FA and FB, respectively, then the point C = (a, c) ∈ Edescribes a closed-trajectory curve kC with areaFC = [aFB + bFA]/(a+ b)+ (c2− ab)h2(t0)piν− 2h2(t0)νcLAB/(∮hdϕ),where LAB is the length of the rod (segment AB) and t0 is a certain time instant during thehomothetic motion.The main theorem of the paper states that if three points A = (0, 0), B = (b, 0) and C = (c, d)∈ E describe closed-trajectory curves of areas FA, FB and FC respectively, then a point X =(x, y) ∈ E describes the areaFX =(1− xb+c− bbdy)FA+(xb− cybd)FB +ydFC+(x2 + y2− bx− c2 + d2dy+bcdy)h2(t0)piν.Reviewed by Philippe A. Mu¨llhauptc© Copyright American Mathematical Society 2008

Review of "Holditch-type theorems under the closed planar homothetic motions"

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Review of "Holditch-type theorems under the closed planar homothetic motions"

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