geometry - How to find the parametric equation of a cycloid? - Mathematics Stack Exchange How to find the parametric equation of a cycloid? up vote 11 down vote favorite 1 "A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia In many calculus books I have, the cycloid, in parametric form, is used in examples to find arc length of parametric equations. This is the parametric equation for the cycloid: $$\begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*}$$ How are these equations found in the first place? 5 Nice picture! Decompose the motion into two parts: (i) steady motion of the centre along the line $y=r$ and (ii) rotation. – André Nicolas Apr 18 '12 at 19:56 I gave a general approach for deriving roulettes (of which your cycloid is a special case) in an answer to this question . – J. M.

Apr 28 '12 at 8:21 Wolfram has quite detailed solutions for not only $r_1 = r_2$ but for all ratios $\frac{r_1}{r_2}$ – Carl Witthoft Jan 21 at 18:23 up vote 5 down vote accepted $t$ measures the angle through which the wheel has rotated, starting with your point in the "down" position. Since the wheel is rolling, the distance it has rolled is the distance along the circumference of the wheel from your point to the "down" position, which (since the wheel has radius $r$) is $rt$. So the centre of the wheel, which was initially at $(0,r)$, is now at $(rt,r)$. Your point is displaced from this by $-r\sin(t)$ horizontally and $-r\cos(t)$ vertically, so it is at $(rt - r\sin(t), r - r\cos(t))$. up vote 2 down vote The center of the circle moves along a horizontal line at constant velocity.

If we want the cusps to be at $y=0$, that means the center should be $(x_c,y_c)=(rt,r)$. Then we add on the location of the point on the rim relative to the center. This will be something like $(r\cos t, r\sin t)$ but we still need to get the phase right. If we start with the point on the rim at $(0,0)$ at $t=0$ the rim point is at an angle of $\frac {-\pi}2$ at $t=0$, that is, pointing straight down. A little fiddling with the phases gets the expression you quote. The scale between the center motion and rotation is set by the requirement that there be no slippage, which means the velocity of the point on the road must be $0$. You should go to the page before reading on and while reading the rest of the post. In it, it explains everything very coherently and breaks down the derivation into 4 steps: finding an equation for the location of the center of the circle (x and y coordinates), and then finding the equation for the point P in in reference to the center.

We will start off by trying to find where the center of the circle is at angle $\theta$. The x coordinate is going to be equal to the distance traveled, which is the same thing as the length of the sector of the circle we have already covered. The sector is equal to the radius times the central angle, so the center will be at $x = a \theta$ The y coordinate of the center at any time is really easy because the center is always the height of the radius, which is $a$. Therefore, the center is at coordinates $(a\theta, a)$ at angle $\theta$. Now, let's try and find the location of point P in reference to the center. We will start with the x coordinate. At angle $\theta$, P will start by lagging behind, then jumping ahead, then going back to where it started. Therefore, we want to start by subtracting $0a$, then $1a$, then $0a$, then -$1a$, then going back to $0$ again.

This behavior is exhibited by $a \sin \theta$, so our x coordinate is now complete: $x = a\theta - a \sin \theta = a(\theta - \sin \theta)$ Now for the y coordinate. To get the height of point P at angle $\theta$, we notice that it starts out below the center, then goes above the center, then back below. Therefore, we want to subtract $1a$, then $0a$, then $-1a$ (add $1a$), then go back to $0a$ again. The pattern of $(1, 0, -1, 0, 1)$ is exhibited by $a \cos \theta$, so we want to subtract this from the center, giving us $y = a - a \cos \theta$ , or $y = a(1 - \cos \theta)$. Now, we are done. Our two equations are $$x = a(\theta - \sin \theta)$$ $$y = a(1 - \cos \theta)$$. cycloid cycloid The cycloid (blue) and its generating circle (red) Types of cycloid The shape defined by a fixed point on a wheel as it rolls.

More precisely, it is the locus of a point on the rim of a circle rolling along a perfectly straight line. The curve resembles a succession of arches, with cusps separated by distances equal to the circumference of the circle. The cycloid was named by Galileo in 1599. It is the solution to both the tautochrone problem and the brachistochrone problem . In 1634, the French mathematician Gilles de Roberval (1610–1675) showed that the area under a cycloid is three times the area of its generating circle. In 1658, the English architect Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle. But there was a lot of bickering and a lack of public sharing of information around this time that led to much duplication of effort, particularly over questions related to the cycloid. In fact the confusion was so bad that the curve was nicknamed the Helen of Geometers, and Jean Montucla referred to it as "la pomme de discorde." The cycloid is of architectural interest since it forms the strongest known arch .

As well as the ordinary cycloid there is the curtate cycloid, which is the path traced out by a point on the inside of a rolling circle, and the prolate cycloid, which is followed by a point on the outside of the circle. A prolate cycloid is traced out, for example, by points on the flange of the wheels of a locomotive, which extends below the top of the tracks. This leads to the surprising conclusion that even as the locomotive is moving forward there are always parts of its wheels that are going backward for a moment before moving forward again. Chapter 7. Gears Figure 7-2 shows two mating gear teeth, in which Tooth profile 1 drives tooth profile 2 by acting at the instantaneous contact point K. N1N2 is the common normal of the two profiles. N1 is the foot of the perpendicular from O1 to N1N2 N2 is the foot of the perpendicular from O2 to N1N2. Figure 7-2 Two gearing tooth profiles Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other.

Therefore, we have (7-1) or (7-2) We notice that the intersection of the tangency N1N2 and the line of center O1O2 is point P, and (7-3) Thus, the relationship between the angular velocities of the driving gear to the driven gear, or velocity ratio, of a pair of mating teeth is (7-4) Point P is very important to the velocity ratio, and it is called the pitch point. Pitch point divides the line between the line of centers and its position decides the velocity ratio of the two teeth. The above expression is the fundamental law of gear-tooth action. 7.2.2 Constant Velocity Ratio For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion transmission between two gears is equivalent to the motion transmission between two imagined slipless cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centers are at O1 and O2, and through pitch point P. These two circle are termed pitch circles. The velocity ratio is equal to the inverse ratio of the diameters of pitch circles. This is the fundamental law of gear-tooth action.

The fundamental law of gear-tooth action may now also be stated as follow (for gears with fixed center distance) (Ham 58) : The common normal to the tooth profiles at the point of contact must always pass through a fixed point (the pitch point) on the line of centers (to get a constant velocity ration). 7.2.3 Conjugate Profiles To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must pass through the corresponding pitch point , which is decided by the velocity ratio. The two profiles which satisfy this requirement are called conjugate profiles. Sometimes, we simply termed the tooth profiles which satisfy the fundamental law of gear-tooth action the conjugate profiles. Although many tooth shapes are possible for which a mating tooth could be designed to satisfy the fundamental law, only two are in general use: the cycloidal and involute profiles. The involute has important advantages -- it is easy to manufacture and the center distance between a pair of involute gears can be varied without changing the velocity ratio . Thus close tolerances between shaft locations are not required when using the involute profile. The most commonly used conjugate tooth curve is the involute curve (Erdman & Sandor 84) .

7.3 Involute Curve The following examples are involute spur gears. We use the word involute because the contour of gear teeth curves inward. Gears have many terminologies, parameters and principles. One of the important concepts is the velocity ratio, which is the ratio of the rotary velocity of the driver gear to that of the driven gears. The SimDesign file for these gears is simdesign/gear15.30.sim. The number of teeth in these gears are 15 and 30, respectively. If the 15-tooth gear is the driving gear and the 30-teeth gear is the driven gear, their velocity ratio is 2. Other examples of gears are in simdesign/gear10.30.sim and simdesign/gear20.30.sim 7.3.1 Generation of the Involute Curve Figure 7-3 Involute curve The curve most commonly used for gear-tooth profiles is the involute of a circle. This involute curve is the path traced by a point on a line as the line rolls without slipping on the circumference of a circle.

It may also be defined as a path traced by the end of a string which is originally wrapped on a circle when the string is unwrapped from the circle. The circle from which the involute is derived is called the base circle. In Figure 7-3 , let line MN roll in the counterclockwise direction on the circumference of a circle without slipping. When the line has reached the position M'N', its original point of tangent A has reached the position K, having traced the involute curve AK during the motion. As the motion continues, the point A will trace the involute curve AKC. 7.3.2 Properties of Involute Curves The distance BK is equal to the arc AB, because link MN rolls without slipping on the circle. For any instant, the instantaneous center of the motion of the line is its point of tangent with the circle. Note: We have not defined the term instantaneous center previously. The instantaneous center or instant center is defined in two ways (Bradford & Guillet 43) : When two bodies have planar relative motion, the instant center is a point on one body about which the other rotates at the instant considered.

When two bodies have planar relative motion, the instant center is the point at which the bodies are relatively at rest at the instant considered. The normal at any point of an involute is tangent to the base circle. Because of the property (2) of the involute curve, the motion of the point that is tracing the involute is perpendicular to the line at any instant, and hence the curve traced will also be perpendicular to the line at any instant. There is no involute curve within the base circle. Figure 7-4 shows some of the terms for gears. Figure 7-4 Spur Gear In the following section, we define many of the terms used in the analysis of spur gears. Some of the terminology has been defined previously but we include them here for completeness. (See (Ham 58) for more details.) Pitch surface : The surface of the imaginary rolling cylinder (cone, etc.) that the toothed gear may be considered to replace. Pitch circle: A right section of the pitch surface. Addendum circle: A circle bounding the ends of the teeth, in a right section of the gear. Root (or dedendum) circle: The circle bounding the spaces between the teeth, in a right section of the gear.

Addendum: The radial distance between the pitch circle and the addendum circle. Dedendum: The radial distance between the pitch circle and the root circle. Clearance: The difference between the dedendum of one gear and the addendum of the mating gear. Face of a tooth: That part of the tooth surface lying outside the pitch surface. Flank of a tooth: The part of the tooth surface lying inside the pitch surface. Circular thickness (also called the tooth thickness) : The thickness of the tooth measured on the pitch circle. It is the length of an arc and not the length of a straight line. Tooth space: The distance between adjacent teeth measured on the pitch circle. Backlash: The difference between the circle thickness of one gear and the tooth space of the mating gear. Circular pitch p: The width of a tooth and a space, measured on the pitch circle. Diametral pitch P: The number of teeth of a gear per inch of its pitch diameter. A toothed gear must have an integral number of teeth. The circular pitch, therefore, equals the pitch circumference divided by the number of teeth. The diametral pitch is, by definition, the number of teeth divided by the pitch diameter.

That is, (7-5) That is, the product of the diametral pitch and the circular pitch equals . Module m: Pitch diameter divided by number of teeth. The pitch diameter is usually specified in inches or millimeters; in the former case the module is the inverse of diametral pitch. Fillet : The small radius that connects the profile of a tooth to the root circle. Pinion: The smaller of any pair of mating gears. The larger of the pair is called simply the gear. Velocity ratio: The ratio of the number of revolutions of the driving (or input) gear to the number of revolutions of the driven (or output) gear, in a unit of time. Pitch point: The point of tangency of the pitch circles of a pair of mating gears. Common tangent: The line tangent to the pitch circle at the pitch point. Line of action: A line normal to a pair of mating tooth profiles at their point of contact. Path of contact: The path traced by the contact point of a pair of tooth profiles. Pressure angle : The angle between the common normal at the point of tooth contact and the common tangent to the pitch circles. It is also the angle between the line of action and the common tangent.

Base circle :An imaginary circle used in involute gearing to generate the involutes that form the tooth profiles. Hi all, I have created three type of Cycloid Curve ,CurtateCycloid ,and ProtateCycloid and Cycloid using GCSCript programming and model based approach. Enjoy watching Cycloid Curve Avi Here is some history about cycloid curve Definition : The shape defined by a fixed point on a wheel as it rolls. More precisely, it is the locus of a point on the rim of a circle rolling along a perfectly straight line. The curve resembles a succession of arches, with cusps separated by distances equal to the circumference of the circle. The cycloid was named by Galileo in 1599. It is the solution to both the tautochrone problem and the brachistochrone problem . In 1634, the French mathematician Gilles de Roberval (1610–1675) showed that the area under a cycloid is three times the area of its generating circle. In 1658, the English architect Christopher Wren showed that the length of a cycloid is four times the diameter of its generating circle.

The above two statement satisfies by the cycloid curve Created GC.the area under the cycloid curve is 84.37 and area of circle with radius 3 is 28.27 which is exactly 4 times of the area of the circle.and Length of the Cycloid Curve is 24 which is exactly four times of the diameter of the circle of radius 3. As well as the ordinary cycloid there is the Curtate cycloid, which is the path traced out by a point on the inside of a rolling circle, and the Prolate cycloid, which is followed by a point on the outside of the circle. A Prolate cycloid is traced out, for example, by points on the flange of the wheels of a locomotive, which extends below the top of the tracks. This leads to the surprising conclusion that even as the locomotive is moving forward there are always parts of its wheels that are going backward for a moment before moving forward again. Cycloids play a major role in gears. They are perfectly meshing (gives perfect rolling action), and earlier was used in manufacturing gears. Mechanical clocks still use cycloidal gears.

Because the contact force direction (pressure angle) changes rapidly for cycloidal gears, however, they are subjected to fatigue stresses and not used in modern gears now a days. Instead involute curve is used for these gears. However, at the extremities of addenda and dedenda of the gears are still manufactured with cycloidal profiles, as involute rapidly changes its shape near these extremities. thanks Gear & GEAR TRAINS |authorSTREAM gear & GEAR TRAINS Does not support media & animations Automatically changes to Flash or non-Flash embed The presentation is successfully added In Your Favorites . Views: This Presentation is Public Favorites: 1 INTERESTING & GOOD FOR BASICS OF GEA Comments By: vallika.sairam (15 month(s) ago) sir plz forward me tis ppt.... my email: vallika.sairam@gmail.com Saving..... By: vallika.sairam (15 month(s) ago) send the ppt of geear measurement to mmy email id rauzsha@gmail.com Saving..... By: dinusha38 (33 month(s) ago) PLZ mail me dear... My email :- dinushachandima38@gmail.com Saving.....

By: dinusha38 (33 month(s) ago) PLZ mail me dear... My email :- dinushachandima38@gmail.com Saving..... By: dinusha38 (33 month(s) ago) PLZ mail me dear... My email :- dinushachandima38@gmail.com Saving..... Kinematics of machines Gear & Gear trains : Kinematics of machines Gear & Gear trains By: Samir Jariwala GEAR…..: GEAR….. Power transmission is the movement of energy from its place of generation to a location where it is applied to performing useful work A gear is a component within a transmission device that transmits rotational force to another gear or device TYPES OF GEARS: TYPES OF GEARS 1. According to the position of axes of the shafts. a. Parallel 1.Spur Gear 2.Helical Gear 3.Rack and Pinion b. Intersecting Bevel Gear c.

Non-intersecting and Non-parallel worm and worm gears SPUR GEAR: SPUR GEAR Teeth is parallel to axis of rotation Transmit power from one shaft to another parallel shaft Used in Electric screwdriver, oscillating sprinkler, windup alarm clock, washing machine and clothes dryer External and Internal spur Gear…: External and Internal spur Gear… Helical Gear: Helical Gear The teeth on helical gears are cut at an angle to the face of the gear This gradual engagement makes helical gears operate much more smoothly and quietly than spur gears One interesting thing about helical gears is that if the angles of the gear teeth are correct, they can be mounted on perpendicular shafts, adjusting the rotation angle by 90 degrees Helical Gear…: Helical Gear… Rack and pinion: Rack and pinion Rack and pinion gears are used to convert rotation (From the pinion) into linear motion (of the rack) A perfect example of this is the steering system on many cars Straight and Spiral Bevel Gears: Straight and Spiral Bevel Gears WORM AND WORM GEAR: WORM AND WORM GEAR PowerPoint Presentation: Forms of Teeth In actual practice following are the two types of teeth commonly used 1.

Cycloidal teeth ; and 2. Involute teeth. Cycloidal Teeth A cycloid is the curve traced by a point on the circumference of a circle which rolls without slipping on a fixed straight line. When a circle rolls without slipping on the outside of a fixed circle, the curve traced by a point on the circumference of a circle is known as epi-cycloid . On the other hand, if a circle rolls without slipping on the inside of a fixed circle, then the curve traced by a point on the circumference of a circle is called hypo-cycloid. PowerPoint Presentation: Construction of cycloidal teeth for gear The cycloidal teeth of a gear may be constructed as shown in Fig. ( b). The circle C is rolled without slipping on the outside of the pitch circle and the point P on the circle C traces epi-cycloid PA, which represents the face of the cycloidal tooth. The circle D is rolled on the inside of pitch circle and the point P on the circle D traces hypo-cycloid PB, which represents the flank of the tooth profile. The profile BPA is one side of the cycloidal tooth. The opposite side of the tooth is traced as explained above.

PowerPoint Presentation: Involute Teeth An involute of a circle is a plane curve generated by a point on a tangent, which rolls on the circle without slipping or by a point on a taut string which in unwrapped from a reel as shown in Fig. In connection with toothed wheels, the circle is known as base circle. The involute is traced as follows : A 3, the tangent A 3 T to the involute is perpendicular to P 3 A 3 and P 3 A 3 is the normal to the involute. In other words, normal at any point of an involute is a tangent to the circle. NOMENCLATURE OF SPUR GEARS: NOMENCLATURE OF SPUR GEARS PowerPoint Presentation: Pitch circle. It is an imaginary circle which by pure rolling action would give the same motion as the actual gear. Pitch circle diameter. It is the diameter of the pitch circle. The size of the gear is usually specified by the pitch circle diameter. It is also known as pitch diameter. Pitch point. It is a common point of contact between two pitch circles. Pitch surface. It is the surface of the rolling discs which the meshing gears have replaced at the pitch circle.

Pressure angle or angle of obliquity. It is the angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point. It is usually denoted by φ . The standard pressure angles are 14 1/2 ° and 20°. PowerPoint Presentation: Addendum. It is the radial distance of a tooth from the pitch circle to the top of the tooth. Dedendum. It is the radial distance of a tooth from the pitch circle to the bottom of the tooth. Addendum circle. It is the circle drawn through the top of the teeth and is concentric with the pitch circle. Dedendum circle. It is the circle drawn through the bottom of the teeth. It is also called root circle. Note : Root circle diameter = Pitch circle diameter × cos φ , where φ is the pressure angle. PowerPoint Presentation: Circular pitch. It is the distance measured on the circumference of the pitch circle from a point of one tooth to the corresponding point on the next tooth. It is usually denoted by P c , Mathematically, A little consideration will show that the two gears will mesh together correctly, if the two wheels have the same circular pitch.

Note : If D 1 and D 2 are the diameters of the two meshing gears having the teeth T 1 and T 2 respectively, then for them to mesh correctly, PowerPoint Presentation: Diametral pitch. It is the ratio of number of teeth to the pitch circle diameter in millimetres. It is denoted by p d . Mathematically, Module. It is the ratio of the pitch circle diameter in millimeters to the number of teeth. It is usually denoted by m. Mathematically, Clearance. It is the radial distance from the top of the tooth to the bottom of the tooth, in a meshing gear. A circle passing through the top of the meshing gear is known as clearance circle. Total depth. It is the radial distance between the addendum and the dedendum circles of a gear. It is equal to the sum of the addendum and dedendum. PowerPoint Presentation: Face of tooth. It is the surface of the gear tooth above the pitch surface. Flank of tooth. It is the surface of the gear tooth below the pitch surface. Top land. It is the surface of the top of the tooth. Face width.

It is the width of the gear tooth measured parallel to its axis. Profile. It is the curve formed by the face and flank of the tooth. PowerPoint Presentation: Comparison Between Involute and Cycloidal Gears In actual practice, the involute gears are more commonly used as compared to cycloidal gears, due to the following advantages : Advantages of involute gears The most important advantage of the involute gears is that the centre distance for a pair of involute gears can be varied within limits without changing the velocity ratio. This is not true for cycloidal gears which requires exact centre distance to be maintained. In involute gears, the pressure angle, from the start of the engagement of teeth to the end of the engagement, remains constant. It is necessary for smooth running and less wear of gears. But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero at pitch point, starts decreasing and again becomes maximum at the end of engagement. This results in less smooth running of gears. The face and flank of involute teeth are generated by a single curve where as in cycloidal gears, double curves ( i.e.

epi-cycloid and hypo-cycloid) are required for the face and flank respectively. Thus the involute teeth are easy to manufacture than cycloidal teeth. In involute system, the basic rack has straight teeth and the same can be cut with simple tools. Note : The only disadvantage of the involute teeth is that the interference occurs with pinions having smaller number of teeth. This may be avoided by altering the heights of addendum and dedendum of the mating teeth or the angle of obliquity of the teeth. PowerPoint Presentation: Advantages of cycloidal gears Following are the advantages of cycloidal gears : Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth. In cycloidal gears, the contact takes place between a convex flank and concave surface, whereas in involute gears, the convex surfaces are in contact. This condition results in less wear in cycloidal gears as compared to involute gears. However the difference in wear is negligible. In cycloidal gears, the interference does not occur at all.

Though there are advantages of cycloidal gears but they are outweighed by the greater simplicity and flexibility of the involute gears. PowerPoint Presentation: Interference in Involute Gears Fig. shows a pinion with centre O 1, in mesh with wheel or gear with centre O 2. MN is the common tangent to the base circles and KL is the path of contact between the two mating teeth. The tip of tooth on the pinion will then undercut the tooth on the wheel at the root and remove part of the involute profile of tooth on the wheel. This effect is known as interference, and occurs when the teeth are being cut. In brief, the phenomenon when the tip of tooth undercuts the root on its mating gear is known as interference. A little consideration will show, that if the radius of the addendum circle of pinion is increased to O 1 N, the point of contact L will move from L to N. When this radius is further increased, the point of contact L will be on the inside of base circle of wheel and not on the involute profile of tooth on wheel.

PowerPoint Presentation: Similarly, if the radius of the addendum circle of the wheel increases beyond O 2 M, then the tip of tooth on wheel will cause interference with the tooth on pinion. The points M and N are called interference points. From the above discussion, we conclude that the interference may only be avoided, if the point of contact between the two teeth is always on the involute profiles of both the teeth. In other words, interference may only be prevented, if the addendum circles of the two mating gears cut the common tangent to the base circles between the points of tangency. When interference is just avoided, the maximum length of path of contact is MN when the maximum addendum circles for pinion and wheel pass through the points of tangency N and M respectively as shown in Fig. Obviously, interference may be avoided if the path of contact does not extend beyond interference points. The limiting value of the radius of the addendum circle of the pinion is O 1 N and of the wheel is O 2 M. PowerPoint Presentation: Measurement of tooth thickness The tooth thickness is generally measured at pitch circle and is therefore, the pitch line thickness of the tooth.

Following method is used for measuring the gear tooth thickness : Measurement of tooth thickness by gear tooth vernier caliper. The gear tooth thickness can be conveniently measured by a gear tooth vernier as shown in the fig. Since the gear tooth thickness varies from the tip to the base circle of the tooth, the instrument must be capable of measuring the tooth thickness at a specified position on the tooth. The gear tooth vernier has two vernier scales. The vertical vernier scale is used to set the depth (d) along the pitch circle from the top surface of the tooth at which the width (w) has to be measured. While the horizontal vernier scale is used to measure the width (w) of the teeth. PowerPoint Presentation: Considering one gear tooth, the theoretical values of w and d can be found out which may be verified by the instrument. As shown in the figure , w is a chord ADB, but tooth thickness is specified as an arc distance AEB. Also the depth d adjusted on the instrument is slightly greater than the addendum CE", width w is therefore called chordal thickness and d is called the chordal addendum.

W=AB=2AD WKT, θ =360/4N, Where N= number of teeth. View More Presentations Cycloids Hypotrochoids and Epitrochoids (an investigation in movement using GSP) As a wheel travels in a straight line, the locus of any point on its circumference will be a familiar curve known as a cycloid (click the movie button for a movie). As a wheel rolls on the inside of a circle, points on the circumference of the wheel trace curves known as hypocylcoids whereas when the wheel rolls on the outside of the circle, epicycloids are generated by points on the circumference of the wheel. Finally the curves traced by points on the interior of the wheel are called hypotrochoids and epitrochoids for the wheel rolling on the inside and outside of the circle respectively. My interest in this paper is to see how we may use Geometer's Sketchpad (GSP) as a tool for exploring the properties of these figures. I claim that students will learn a great deal more about the figures by writing GSP scripts to draw them than they will from either a lecture or a demonstration.

Let me outline what I anticipate a learning experience may entail: 1. Understanding the Problem GSP has no function whereby you can tell it to "roll the wheel" about a circle. The first task then is to find a manner in which this can be achieved and to do this the student will have to truly understand the problem. That is we need to simulate in some manner a wheel rolling about a circle - since the rolling wheel touches the circle as it rolls, it seems clear that as the wheel completes one revolution it should have touched a fraction of the circle equal to the circumference of the wheel. In this figure the rolling wheel has a radius one third of the circle and after one revolution of the wheel it has "touched" one third of the circle. It also important to notice that the radius of the wheel has gone through 360 degrees but the angle between the starting point on the circle and this point of contact is 360/3 = 120 degrees! 2. The Construction Now that we understand the problem we may begin with the construction.

The key to the construction is the fact that we can both animate two aspects of a GSP sketch simultaneously and that we can be sure that the point moving in one animation does so at the same speed as the point in the other animation (provided that we chose the same setting (quickly/normal/slowly)). The basis of the construction is as follows: Given a circle center O and radius OA, construct a point T on the circumference of the circle. The wheel is constructed on the radius OT with center B'. The easiest would now be to animate T around circle center O while animating some point (S') around the wheel - GSP does not allow this (since the wheel is a result of the point T and you cannot animate on something that is the result of another animation) and we must introduce a second circle. Hence the construction: Given OA and the circle (center O passing through A) construct another line O'A' = OA and some point B on the line segment O'A' . The wheel is constructed on radius OT so that O'B = B'T where T is an arbitrary point on the circle. S is an arbitrary point on the image of the wheel (circle center O' and passing through B).

Finally we construct S' so that angle BO'S = angle TB'S' and if we now animate T around the circle and S around the image of the wheel we get the desired curve by tracing the point S'. This construction provides the basis of all the different curves we wish to investigate...... HOWEVER......... The construction has a small problem with accuracy (which is sadly a function of the software). There are certain properties of these curves that are well known namely: In the case of the hypocylcoids (and epicylcoids) if R/r is the ratio of the radius of the circle to the radius of the wheel and R/r is rational then R will represent the number of cusps while r indicates the number of times the wheel must roll around the circle before the picture is completed (in the figures below R/r = 3, 4 and 2 respectively) Using our construction sadly results in some inaccuracies...

(for discussion of these inaccuracies click here to read some correspondence from the developer) A different construction If we return to our original remarks we we recall (and if it was not obvious then - it should be after the first construction) that there is a relationship between the angle through which the wheel has turned and the angle of the circle through which it has moved this can be given by: angle TBS = R/r * angle TOA (and for complete accuracy: angle TBS = - (R/r * angle TOA)). Using this technique clearly overcomes the problem of inaccuracy described earlier. Sadly, however, it introduces another: In the case where R/r is rational and r is not 1 we get (in the case R/r = 3/2): Notice that the locus is incomplete!

This is easily understood by considering what is actually happening with our construction: In the first case (with R/r = 3) as angle TOA approaches 180 degrees - say at 179 degrees angle TBS = - (3 * 179) = - 537 degrees, then as angle TOA passes 180 degrees to say 181 degrees GSP measures the angle as - 179 degrees and TBS = - (3 * - 179) = 537 degrees. Fortunately the angles - (3 * 181) = 573 and 537 are really the same resultant movement of BS and the sketch is not influenced.

In the second case (with R/r = 3/2) is we consider the same scenario then as TOA changes from 178 degrees to 182 (= - 178) degrees, TBS changes from - 267 degrees (- (3/2 * 178) = - 267) to (- (3/2 * -178) = 267 degrees and this time 267 degrees and -267 degrees do give the same resultant movement of BS and hence the new problem! COMPARISON OF CONSTRUCTIONS Each construction has some advantages and hence some value. The greatest advantage of the second construction is that we can construct the hypocycloid as a locus which enables us to observe the impact of changing R/r without having to animate the curve each time.

Click the button for a Quicktime movie that illustrates this remark: One important observation that you should have made when watching the movie was that the same error that causes the problem when r in R/r is not 1 also causes trouble in this situation - compare the following clip from the movie with the corresponding scene using our other construction (R/r = 7/2 in each case): So one is left with the choice - use the first construction despite its inherent flaw (and increase the accuracy by making some adjustments to the actual dimensions of R for each case) OR use the second construction which is technically flawed but gives a more accurate illusion within certain parameters (R/r has r = 1). 3. The Exploration We are now ready to explore the properties of Cycloids, Hypocycloids, Epicycloids, Hypotrochoids and Epitrochoids For the purpose of this paper I will only provide illustrations (both figures and movies) and GSP sketches (for the reader to play with).

To access the GSP sketch used to draw these hypocycloids click the GSP button: R/r = 2: The diameter of the wheel traces the astroid: R/r = 4: R/r = 4/3 In the case R/r = 4/3 we get exactly the same hypocycloid as in R/r = 4 (though it is not completely clear in the sketch) but the trace of the diameter of the wheel is quite different! To access the GSP sketch used to draw these epicycloids click the GSP button: R/r = 1 - This epicycloid is called a Cardioid The Cardioid may also be considered by using some arbitrary points on the circumference of the fixed circle as centers of circles passing through the cusp of the cardioid as in the figure above.

We can further think of the cardioid as being created by a circle rolling around the fixed circle and having twice the radius of the fixed circle (consider the hypocycloid construction with R/r = 1/2) HYPOTROCHOIDS To access the GSP sketch used to draw these hypotrochoids click the GSP button: R/r = 2, r/f = 2 Rolling Wheel Animation - Peltier Tech Blog Peltier Tech Blog Peltier Technical Services, Inc., Copyright © 2016 . A college science professor named Roger Blickensderfer wrote to me about an animation he was working on. He wanted to trace the path followed by a point on the rim of a rolling wheel. Roger had cobbled something together, and I fiddled with it, and it kind of worked. But I decided I wanted it to do a bit more. I started with Roger’s basic idea and added my flexible animation from the Gas Prices – Animated Bar Chart for Excel example. I added a second point, which can be placed at a different distance from the axis of the wheel, and I added a checkbox to show or hide the curves traced by these two points.

I built an animation which can be extracted from this downloadable zip file . The top of the worksheet has the playback controls and other inputs to the left and the chart to the right. The first example uses a wheel radius of 10 and a second point at a radius of 0 (the axis of the wheel) Press the button to start the animation. The red line stays at the same height, while the blue line traces a curve called a cycloid. Here are the traces when the second point is at a radius of 5 (midradius). Here are the traces when the second point is at a radius of 15 (outside the radius of the wheel rim). This is similar to the case of a train wheel, which has a flange to keep the tain on the track. The outer surface of the flange actually moves backwards at the bottom of its travel. In this example, the second point has a radius of -10, so it appears on the rim opposite the first point. I found another explanation of this in Rolling: the physics of wheels , on the web site of the School of Physics at The University of New South Wales. There is a large collection of similar animations on their Physclips page .

Question: What is the name of the type of curve traced by a point on a rolling wheel?
Answer: Cycloid