Introduction to Trochoids Suppose that one of Ezekiel's wheels is fixed, while the other wheel rolls around its rim. Then a point attached to the rolling wheel traces a curve, called a trochoid. There are a great variety of trochoids. Some look like stars, some like flowers; some, maybe, like both. Along with beauty, they have interesting geometrical properties. The rolling wheel may be inside or outside the static, or supporting, wheel. If it is inside, the curve is called a hypotrochoid; if outside, an epitrochoid. The tracing point may be inside the rolling circle, outside it, or on the circle. If it is inside, then the curve is said to be curtate; if outside, it is prolate. If the point is on the circle, then the curve is an epicycloid or hypocycloid. The word cycloid is usually applied to the curve generated by a point on the rim of a circle that rolls on a straight line. Following the precedent set by F. Morley, I shall use this word to include epicycloids and hypocycloids. A trochoid is a closed curve, of finite length, precisely when the radius of the rolling circle is a rational multiple of the radius of the supporting circle. I will use the convention that this ratio, which I will call the wheel ratio, is positive if the two circles curve the same way at the point of contact. Thus the curve is a hypotrochoid if the wheel ratio is between 0 and 1; if it is negative or greater than 1, then the curve is an epitrochoid. The wheel ratio is w= ± N/D where N and D are positive integers; usually we shall keep the fraction in lowest terms, so that the greatest common divisor of N and D is 1. Then, if the curve is a cycloid (epi- or hypo-), it will have D sharp points, or cusps, on the supporting circle. D is sometimes called the degree of the trochoid. The distance from the center of the rolling circle to the tracing point I will call the arm, and the ratio of this to the radius of that circle I will denote by ρ, and call the arm ratio. When N > 1, the trochoid can be generated by any one of N points equally spaced around the rolling circle. More generally, one can take any number, say M, of points equally spaced around the rolling circle, and ask what curves they generate. Let K be the greatest common divisor of N and M, with M = Km; then these points generate a rosette of m trochoids, each one traced by K points. An Interactive Display of Trochoids and Rosettes Here is* an interactive display of trochoids, or rosettes of trochoids, along with the wheels that generate the curves. The top controls on the display enable you to select the sign, denominator, and numerator of the wheel ratio. Below these is a slider, which gives a range of possible values for the arm ratio, ρ; the selected value is shown below the slider. The interesting special case of cycloids can be chosen by clicking the button labeled "Cycloid." The control labeled "M" enables you to select the number of tracing points. Note that if M is equal to N, then all the points trace out the same trochoid. There is a knob at the center of the rolling circle, represented by a black dot. If you move the cursor to that dot and press the mouse button, the dot will turn red, and it can be moved by dragging the mouse. This shows how the rolling wheel actually generates the trochoid. The dot will turn black, and revert to immobility, if you release the mouse button. Wankel rotary-engine We can partially represent the inner surface of the housing and the movement of the rotor apexes of a Wankel rotary engine by using these values for the controls in the interactive display which should display to the right of this webpage, assuming your browser is able and set to execute the Javascript code. Sign = +

D = 2

N = M = 3

ρ ≅ 2.67

Use the 'Wankel' button I've added to the interactive display to set these values quickly. Admittedly, only the positions of the apexes of the rotor are adequately represented here by the ends of the 3 radial arms of the rolling circle, not how the somewhat triangular rotor shapes the engine chambers. As demonstrated by Tony Kelman's "Wankel Rotary Engine: Epitrochoidal Envelopes", it should be possible to implement the rotor shape as well in a future version of this page. The trochoid constant 'K' In "Rotary Engine" by Kenichi Yamamoto, on pages 20 - 23, Yamamoto defines 'the distance (e) between the center of base circle A and that of the revolving circle B, is referred to as "eccentricity". The arm length R on the revolving circle B is referred to as the "generating radius" of the peritrochoid*."' and in equation 2.11 Yamamoto defines 'K = R/e' where 'K is able to represent the trochoidal configuration, which is known as the trochoid constant'.

We can equate Henrich's arm ratio 'ρ' with Yamamoto's trochoid constant 'K' for any trochoid (where N ≠ D) with K = ρ * N / ( |N - D| ) and specifically for the Wankel rotary engine with N=3 and D=2, this simplifies to

K = 3 ρ Yamamoto states 'a value between 6 and 10 is selected for K' which equates to ρ between 2 and 3.34 * Peritrochoid - a trochoid generated using a rolling circle which is larger than the supporting circle, rolling hula-hoop style; mathematically equivalent to a epitrochoid. The eccentricity ratio 'er' Tony Kelman's "Wankel Rotary Engine: Epitrochoidal Envelopes" demonstration features an interactive display with a slider labelled 'eccentricity ratio' which controls the value of the 'er' variable in his source code. For the Wankel rotary-engine trochoids, we can equate Kelman's eccentricity ratio 'er' with Yamamoto's trochoid constant 'K' and Henrich's arm ratio 'ρ' with K er = 1

3 ρ er = 1 Animation To animate the display, use the 'Animate' button I've added to rotate the rolling circle and the same button stops the animation. The direction of rotation alternates with successive button presses between clockwise and anti-clockwise. Follow-mouse-cursor button-free latch Another option for mouse control of the rolling circle, useful to ease the repetitive strain of having to hold the mouse button down is featured and documented here. Latch the display in follow-mouse-cursor button-free mode by firstly turning the control knob red as before by clicking and holding down the mouse cursor on the rolling circle's centre dot but now drag the cursor into the Indian Red border surrounding the display window and then release the mouse button. This latches the interactive display in follow-mouse-cursor button-free mode. To unlatch simply click inside the display window. Algebraic Representation We can find an expression for the coordinates of an arbitrary point on a given trochoid. More Double Generation of a Trochoid Double Generation Theorem: If a trochoid is generated with a supporting circle of radius R , wheel ratio w , and arm ratio ρ , then the same trochoid can also be generated with a supporting circle of radius R′ , wheel ratio w′ , and arm ratio ρ′ , where R′ = ρR, w′ = 1 - w, ρ′ = 1/ρ Here is an interactive display which demonstrates double generation. The controls are like those of the display farther up on this page; their values refer to the blue circles and arms. The values of R′, D′, N′, and ρ′ refer to the green circles and arms. A peritrochoid is an epitrochoid The Wankel button option in this interactive display illustrates that Yamamoto's peritrochoid generated using D=2 and N=3 and sign = +, is not merely "substantially the same" but mathematically equivalent to the epitrochoid generated using D=2, N=1 and sign= -. Rotary engine rotor-contour equations Tony Kelman in his "Wankel Rotary Engine: Epitrochoidal Envelopes" demonstration describes the curve representing the contour of the rotor as an "envelope" and that's likely to be because he based his Mathematica program code which draws that rotor "envelope" curve on an "inner envelope" equation which he has referenced as derived in a book by R. F. Ansdale, "The Wankel RC Engine: Design and Performance", published in New York: by A. S. Barnes in 1969. According to several reviews I've read, Ansdale's book is a valuable reference for the mathematics of the Wankel rotary engine and other rotary machines but, somewhat frustratingly, unlike Yamamoto's "Rotary Engine" book which one can download as a pdf file for free, Ansdale's book is not yet freely available for download on the internet, so most of what Ansdale writes there remains out of reach to me to me at this time. If anyone out there has a copy of Ansdale within reach and is willing to scan in certain pages of that book which I am keen to read and then email the images to me, I'd be grateful for the help. Wanted: Image scans of Ansdale, pages 136 to 139 plus the part of the appendix which Tony Kelman writes about in this quote "The appendix of the Ansdale book has the equations derived for arbitrary numbers of lobes, if I remember correctly." So that would be most relevant information to help me develop this webpage and so if anyone has hands on access to a copy of Ansdale then please scan and email those pages to me at peterdow@talk21.com. Thanks! Inner envelope equation for the 2-lobe peritrochoid Tony Kelman was kind enough to take the time to help me to compare the rotor envelope equation he had used with the equations which Yamamoto had published for the same thing. I say "equations", plural, because we noticed a critical typographical mistake in Yamamoto's 1971 edition, misstating an inaccurate envelope equation - which typo Yamamoto had corrected in his 1981 edition of "Rotary Engine", as described in this "Compare the TYPO!" image. Equivalence of Ansdale-1969 & Yamamoto-1981 inner envelope equations At first glance it may not look like it but mathematically the following two equations - Ansdale's and Yamamoto's - are indeed equivalent - they both draw exactly the same curve when plotted and give numerically identical values to the available precision. The equivalence of these equations is analytically demonstrated below by replacing the coloured terms of say Ansdale's equation with their similarly coloured equivalent terms in Yamamoto's equation, by rearranging the terms or by using standard trigonometric product-to-sum identities, listed below. Ansdale - 1969

Adjust some of the terms in Ansdale's equation by multiplying by 1 expressed as "½ × 2" and arrange the factors to allow the identification of equivalent factors in both equations. X = R × ( cos 2v

- 3 (e/R)2 × ½ × 2 sin 6v sin 2v

+ (e/R) √ ( 1 - 9 (e/R) 2 sin 2 3v ) × 2 cos 3v cos 2v

) Y = R × ( sin 2v

+ 3 (e/R)2 × ½ × 2 sin 6v cos 2v

+ (e/R) √ ( 1 - 9 (e/R) 2 sin 2 3v ) × 2 cos 3v sin 2v

) Yamamoto - 1981

Adjust some of the terms in Yamamoto's equation by multiplying some terms by 1 expressed as "R × (1/R)" and then express the equation as terms which are factors each of which is multiplied by the common factor of R. Also, rearrange the 2nd X term which Yamamoto has as an addition term with a "(cos 8v - cos 4v)" trig factor by multiplying the term by 1 performed as multiplication by "-1 × -1", changing the addition term to a minus term and swapping the trig factor subtraction minuend and subtrahend noting that "-1 × (cos 8v - cos 4v) = (cos 4v - cos 8v)" . X = R × ( cos 2v

- (1/R) (3e2/2R) × (cos 4v - cos 8v)

+ (1/R) e √ ( 1 - 9 (e/R) 2 sin 2 3v ) × (cos 5v + cos v)

) Y = R × ( sin 2v

+ (1/R) (3e2/2R) × (sin 8v + sin 4v)

+ (1/R) e √ ( 1 - 9 (e/R) 2 sin 2 3v ) × (sin 5v - sin v)

) Trigonometric product-to-sum identities

Use the appropriately colour-matched trigonometric identity to confirm that the trigonometric terms in Ansdale's and Yamamoto's equations are identical. 2 sin A sin B = cos (A-B) - cos(A+B) 2 cos A cos B = cos (A+B) + cos(A-B) 2 sin A cos B = sin (A+B) + sin(A-B) 2 cos A sin B = sin (A+B) - sin(A-B) Mathematica confirmation of Ansdale = Yamamoto by Tony Kelman Tony Kelman has used Mathematica graphically to plot and analytically to confirm the equivalence of the Ansdale and Yamamoto envelope equations and he has kindly sent me the yamamoto-vs-ansdale Wolfram .CDF file which I am publishing at this link. More pages from Christopher J. Henrich on Trochoids Visit the Interactive Math website at MathInteract.com then click on "Trochoids" in the navigation menu to reveal a sub-menu to such pages as Normals And Tangents

Envelopes

Evolutes

Rolling Trochoid

Conformal Maps