The answer to a question on involute gear curve calculations shows how the gear tooth curve proceeds as an involute calculation from the base diameter.

If the pitch diameter is D = m.N where N is the number of teeth and m is the modulus (for metric gears), then the base circle diameter for a pressure angle α, most usually 20 degrees, is D.cos α and the root diameter is given by the amount the tooth falls below the pitch circle. This amount is the dedendum and is usually 1.25m. Thus the root circle diameter is D - 2.5m.

The involute is calculated in a standard way and looks like the unwinding of a string from a circle, the base circle:

enter image description here

The gear tooth is then described by the intersection of the involute with the root diameter (the bottom of the tooth) and the outer diameter (the top of the tooth). An issue arises if the root diameter is less than the base diameter as the involute is not defined below the base diameter (you get a math domain error in Python if you try it!).

enter image description here

These tooth are described as "undercut". I cannot find a description of how the curve proceeds once you "run out of involute" - ie cross the base diameter.

Is there a standard way of describing the rest of the gear tooth curve?

  • $\begingroup$ Are you using a polar formulation for the involute? As the cartesian formulation has no problem with negative distances... Anyway the undercut is not actually part of the (same) involute as it deals with the opposing edge not the toothprofile. $\endgroup$
    – joojaa
    Commented Feb 20, 2017 at 16:00
  • $\begingroup$ Yes I am. In my program I am stepping the distance to the involute (Ob in the first diagram) by a fixed amount each time (up to the outside diameter), using the information to calculate the Involute angle (inv α), and then converting from polar to rectangular. (I'm referencing "Elements of metric gear technology"). I didn't really try to go negative as I just assumed that I couldn't formulate the involute below the base diameter! I'll go get some paper and have another think about the calculations; perhaps I gave in to quickly... $\endgroup$
    – carveone
    Commented Feb 20, 2017 at 22:04

1 Answer 1


In general, each tooth of an involute gear is made up of two curves: the involute curve and the root fillet. The involute curve is essential for gears to operate properly, and allows power to be smoothly transmitted from one gear to the other. The root fillet on the other hand is not involved directly with power transmission or the kinematics of the gears (i.e. how the motion of one gear influences the motions of a meshing gear.). This means that the root fillet curve's shape is reasonably arbitrary, provided, of course, that the root fillet on one gear does not at any point make contact with the teeth of the other gear. In addition, the root fillet also plays a roll in determining how much bending stress develops at the root of the teeth during operation: if the root fillet has a low radius of curvature, this will result in a large stress concentration at the root.

Despite the root fillet curve having the potential to be almost any shape, there are a few shapes that occur commonly, the two which I believe to be most common are as below:

1. Connecting involute curve and root circle with radial lines

The simplest root fillet to determine mathematically is simply a straight radial line that connects the start of the involute curve to the root circle. This is shown below:

enter image description here

The advantages of the root "fillet" (a sharp corner is not much of a fillet I guess) is the geometry is very simple. However, there are two big disadvantages: 1. The sharp corner produces a very large stress concentration at the root of the gear, meaning larger root bending stresses. This means that these gears are just not suitable for applications that subject them to strong torques. 2. In cases where undercut interference becomes an issue, these radial lines are going to get in the way of the teeth, causing gears to jam. For gears with pressure angles at the standard value of 20°, undercut interference usually occurs for gears with teeth less than 18 (if gear is meshing with a rack gear) or 13 (if gear is meshing with another gear of equal number of teeth). For these reasons, you will rarely see gears with this kind of root fillet in industrial application, but are typically fine in areas where teeth numbers are quite high and the gears are not subjected to high torques.

2. Trochoidal root fillet

One of the most common processes for generating spur gears and helical gears is hobbing, and the root fillet that is naturally produced as a result of hobbing is the trochoidal curve. An example of a gear with trochoidal root fillets is shown below:

enter image description here

The individual trochoidal curve and involute curve are displayed along with the root circle and outer circle as below:

enter image description here

The first thing to notice is that the trochoidal fillet is considerably more rounded than the previous case, so less bending stress will develop at the root of the tooth and so the gear is more suitable in high-torque scenarios. Also, notice how the trochoidal curve intersects with the involute curve at a point slightly to the right of the start of the involute: some of the involute curve has been inadvertently lost by the hobbing process! The smaller the number of teeth, the greater this involute loss will be, and it becomes apparent as undercut interference starts to occur: in such a case, the gears are said to be undercut. While for radial-line fillet curves the undercut interference would have caused the gears to jam, gears with trochoidal fillets will not jam in such circumstances.

Whenever gear teeth are generated by hobbing, it is as though a rack gear is rolling along a gear blank (circle with diameter equal to outer diameter of gear tooth), material being subtracted wherever the rack gear has been. The below animation should illustrate how the rack gear can generate the shape of a spur gear with trochoidal fillets:

enter image description here

Notice how the teeth on the rack gear are trapezia shapes (and also that the rack gear here, being a cutter, has an addendum of $1.25m$ and a dedendum of $m$, unlike usual rack gears), and the cutting of these trapezium teeth into a gear blank manages to magically create two curves, the involute and the trochoidal curves. More precisely, the involute curve is generated by either of the diagonal lines of the trapezium, and the trochoidal curve is generated by either of the two points on the end of the trapezium. With this information, it is possible to express the trochoidal fillet as a parametric equation.

The next section of this question will look at the mathematics behind deriving such a parametric equation.

Deriving the trochoidal curve

First of all, let's consider a single trapezium of the rack gear, and see how a point on the end of a trapezium moves relative to the gear being cut. The curve that this point follows will be the trochoidal curve. The pitch point of the trapezium, $P$, is the point on the trapezium that will initially lie on the gear's pitch circle, as shown below.

enter image description here

Note that the pitch circle radius is $R$, and that there is a general point $Q$ that lies on the trapezium, offset from $P$ by $x_0$ and $y_0$ in the $x$ and $y$ directions (later, we will set this point to be one of the points on the end of the trapezium). Also, note the solid black triangle: this is to help identify the orientation of the gear. The current position of $Q$ can be expressed as follows:

$$ \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} R+x_0 \\ y_0 \end{array} \right] $$

Now, let's rotate the gear anticlockwise by an angle $t$, which then pushes the trapezium up by distance $Rt$ (no slip between pitch circle and pitch line). Illustration:

enter image description here

Note how the solid black triangle has moved in accordance with the gear rotation. The position of $Q$ is now:

$$ \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} R+x_0 \\ Rt+y_0 \end{array} \right] $$

The trapezium has moved, but so has the gear: we are interested in the motion of $Q$ relative to the gear. Therefore, we need to rotate the entire system about the origin by angle $t$ clockwise. This simply rotates the perspective so that it returns the gear to its default position while retaining the new movement of the trapezium, i.e. we get the motion of a trapezium as the rack rolls around the stationary gear, similar to the animation above.

enter image description here

To get the position of point $Q$, we need to multiply the previous position by a clockwise rotation matrix, so the position is expressed as follows:

$$ \left[ \begin{array}{c} x \\ y \end{array} \right] = \begin{bmatrix} \cos{t} & \sin{t} \\ -\sin{t} & \cos{t} \end{bmatrix}\left[ \begin{array}{c} R+x_0 \\ Rt+y_0 \end{array} \right]= \left[ \begin{array}{c} (R+x_0)\cos{t}+(Rt+y_0)\sin{t} \\ -(R+x_0)\sin{t}+(Rt+y_0)\cos{t} \end{array} \right] $$

Finally, by setting $Q$ to be the "upper left" point on the trapezium, i.e. $x_0 = -1.25m$ and $y_0 = \frac{1}{4}\pi m - 1.25m\tan{\phi}$, where $\phi$ is pressure angle, and also noting that $R=\frac{1}{2}Nm$, where $N$ is number of gear teeth, you get the parametric equation for the trochoidal fillet curve:

$$ \left[ \begin{array}{c} x(t) \\ y(t) \end{array} \right] = \left[ \begin{array}{c} (\frac{1}{2}Nm-1.25m)\cos{t}+(\frac{1}{2}Nmt+\frac{1}{4}\pi m - 1.25m\tan{\phi})\sin{t} \\ -(\frac{1}{2}Nm-1.25m)\sin{t}+(\frac{1}{2}Nmt+\frac{1}{4}\pi m - 1.25m\tan{\phi})\cos{t} \end{array} \right] $$

Note that the parameter $t$ will start at $t=-\frac{y_0}{R}=-\frac{\frac{1}{4}\pi - 1.25\tan{\phi}}{\frac{1}{2}N}$ and increases up until a particular value that represents the intersection of the trochoid with the involute.

Hope that answers your question :)

  • 4
    $\begingroup$ Whoa! That's a very nice explanation of the trochoidal curve. Actually the only proper explanation I've seen so far :-) I've seen the hobbing process in action cutting a gear but there's a bit of leap from there to describing the process mathematically! What seeing it did tell me is that you can change the gear to be less undercut by simply shifting the hob cut depth out a bit (profile shifting) which can help when cutting gears with few teeth. Takes a little experimentation sometimes to find out what depth change is useful though... $\endgroup$
    – carveone
    Commented Feb 21, 2017 at 14:16
  • 1
    $\begingroup$ What i have never understood is why most of the literature strive to describe circle involute and torchoid curves in polar form. These are so much easer to understand and think out in parametric equations in cartesian coordinates. $\endgroup$
    – joojaa
    Commented Feb 21, 2017 at 15:00
  • 1
    $\begingroup$ Also its not like you can not round the bottom corner of the sraight line extension ;) $\endgroup$
    – joojaa
    Commented Feb 21, 2017 at 15:05
  • $\begingroup$ That is certainly true! Forgot about that :) I'll edit that into the answer once I have a moment. $\endgroup$
    – Involute
    Commented Feb 21, 2017 at 15:08
  • $\begingroup$ That's true - in the cad program I am using I can just add a fillet. But if the cases of undercut interference is such that radial lines get in the way of the teeth, adding fillets would make that even worse. But from looking at the source code of one of the examples in Autodesk's Fusion 360, that what they do - radial line, add fillet. $\endgroup$
    – carveone
    Commented Feb 21, 2017 at 16:08

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