Rocker Radius Gauge [Gnurd Alert: Mathematical Discussion]

[tstop4me]

Note:  I've spun the present thread from this previous thread:  https://skatingforums.com/index.php?topic=8715.0., starting with Reply #27.  The topic of the previous thread was a low-cost, basic even edge checker; the thread then went way off-topic with discussion of rocker radius gauges.

Gnurd Alert:  This thread discusses a mathematical issue, raised in the previous thread, with rocker radius gauges.  It will not be of interest to general skaters; however, we do have members with a scientific or engineering background.

I.  Rocker Radius Gauges:  Principle of Operation

* A rocker radius gauge operates on the mathematical principle that the locations of three known points on the circumference of a circle determine the center and the radius of the circle.  In a typical rocker radius gauge, two of the points are established by contact between two guide pins and the rocker of the blade.  The third point is established by contact between the tip of a dial indicator, located in between the two guide pins, and the rocker of the blade.

* The Skateology Rocker Radius Gauge, a dedicated rocker radius gauge, is described here:  http://www.iceskateology.com/Skateology/Rocker_Radius_Gauge.html. It is a symmetrical gauge, in that the dial indicator is positioned midway between the two guide pins (called index buttons on the webpage).  The symmetrical configuration leads to the simplified geometry shown in PIX. 1.  The known quantities are the chord length 2L (distance between the contact points of the two guide pins) and the arc height H, which is measured by the dial indicator.  Note: The zero value of the dial indicator is set by applying the rocker radius gauge to a straight edge.  The analysis is straightforward by plane geometry and trigonometry; for trigonometry, no complexity beyond the Pythagorean Theorem is needed.  Equation [E3] in PIX. 1 shows the calculation of the rocker radius from the values of the chord length and arc height. 

The webpage mentions that in the original version of their rocker radius gauge, the reading on the dial indicator was converted to radius via a supplied graph.  In their current version, the reading gives the radius directly.

* The Edge Specialties/Wissota Hollow Depth Indicator (HDI) Gauge is described here:  https://wissota.com/product/hollow-depth-indicator-h-d-i/.  It is primarily designed to measure the evenness of edges and the depth of hollow (DOH) of a blade.  From the DOH and the thickness of the blade (measured separately by a micrometer or caliper), the radius of hollow (ROH) can be calculated.  Incidentally, the HDI Gauge can be used to determine rocker radius, as claimed on the webpage (however, no details are given in the instruction manual):  it contains the same key components of the Skateology Rocker Radius Gauge.  Since the dial indicator can be moved transversely from inside edge to outside edge, however, the dial indicator must be properly positioned on a suitable edge for rocker radius measurements, as previously noted by Kaitsu:

You need to ensure also that dial gauge is in the highest point of the edge and not either side of edge highest point. Dial gauge tip and pins has to have same contact line.

In contrast to the Skateology Rocker Radius Gauge, the HDI Gauge is an asymmetrical gauge, in that the dial indicator is not positioned midway between the two guide pins.  PIX 2.  shows a partially disassembled HDI Gauge applied to a blade.  Two guide pins and a dial indicator tip are in contact with the blade.

PIX. 3 (blade absent) shows details of the partially disassembled gauge.  The dial indicator tip is clearly not positioned midway between the two guide pins.  Solving for the rocker radius from (a) the known distances between the contact point of the dial indicator tip and the contact points of the two guide pins and (b) the arc height, as measured by the dial indicator, is messy if approached via plane geometry and trigonometry, as previously noted by Kaitsu:

Note that in H.D.I gauge dial gauge is not positioned symmetrically in the middle of the pins like in Sid gauge. In my opinion you will need a bit more complex calculations to get rocker radius even you would have described straight calibration piece.

But the solution is fairly straightforward if we use analytical geometry and choose the reference frame wisely.  PIX. 3 shows a Cartesian coordinate reference frame, with the X-axis passing through the contact points of Guide Pin 1 and Guide Pin 2, and the Y-axis (perpendicular to the X-axis) passing through the contact point of Guide Pin 1. 

PIX. 4 then shows the coordinates of three known points on the circle:  the contact point of Guide Pin 1, the contact point of Guide Pin 2, and the contact point of the dial indicator tip.  Note: The zero value of the dial indicator is set by applying the HDI Gauge to a straight edge.  Equation [E4] in PIX. 4, gives the general equation of a circle with the center at coordinates (x_c, y_c) and the radius r.  Expression [E5] gives the sets of coordinates of the three known points.  If we enter the values of the three sets of coordinates in [E4], we generate a system of three equations with three unknowns (x_c, y_c, r)  .  The solution of this system of equations is then given in [E6] – [E8]. 

As a first sanity check, we note from [E6] that  x_c =x₁/2.  That is, the center of  the circle lies along the perpendicular bisector of the chord  CH(P₀-P₁), as expected.

As a second sanity check, we consider the instance of a symmetrical radius gauge with  x₂ = x₁/2.  If we substitute x₁ = 2L and y₂ = H, then [E5] – [E8] do yield a radius  r in agreement with [E3] in PIX. 1.


* If the HDI Gauge as supplied can indeed function as a suitable rocker radius gauge, then owners of this gauge can be spared the purchase of a separate dedicated rocker radius gauge.  In a future post, we will discuss the limitations of the HDI Gauge for determining rocker radius.

[Click Images to Enlarge]

[Kaitsu]

Very impressive calculation formulas and pictures. I really appreciate those whom has real knowledge....no matter if are talking about the calculation formulas or any other topics. There are some people whom think they know everything from everything even in the reality they do not know so much about the discussed topic. I might do this also by my selves from time to time.

Back to actual topic...
One another major difference between the H.D.I gauge and Rocker Radius Gauge is the pins distance and where the pins take contact to the blade. In H.D.I gauge pins are quite far from the each others and this may cause some problems when you get closer to area where spinning rocker begins. You can read more about the idea of index pins positioning from the link what tstop4me shared.

H.D.I gauge cannot be either slid along the blade as pins lays against the edges...at leas if blade is freshly sharpened. Rocker Radius Gauge takes contact from the hollow so it has lower risk to damage edges. Personally I would like to measure rocker radius from the edges which are really used and not from the hollow, but without damaging the edges.

Both gauges are for me more like a big boy toys. I use them mainly to demonstrate things for the people. All other time they are just laying as a man gave decor.

[supersharp]

Thanks for the pin comparison photo. 

I use the rocker radius gauge on new blades to see how consistent the rocker is. I don't have the HDI, but I have plenty of other expensive man-made wall decorations, which I think is inevitable.  It's hard to know which tool is really going to be the magic one until it is in your hands and you get a chance to use it a bit.

I love clever tools but I think I will not invest in the HDI...