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Sine Wave Rosette and Reciprocation Patterns on the End of a Piece

by Roy Lindley

The ornamental turner has many options for surface embellishments. A common one is the simple sine wave or sinusoidal like forms that closely match a theoretical mathematical form. On a lathe these can be achieved mechanically with rocking or pumping rosettes or a reciprocator mechanism. Cutter translation along the cylinder axis with spindle reciprocation produces sinusoids on the cylindrical surface or a distorted sinusoid on the face of a workpiece. A sinusoidal pumping rosette produces the pattern around the cylinder perimeter and the rocking rosette creates the sinusoid about a circle on the face of the workpiece. Cylinder surface wave forms, whether from pumping or reciprocation, closely resemble a true sine pattern but those on the face of a workpiece are distorted uniquely because of radius effects. This article compares resulting workpiece face patterns for the traditional rocking wave forms as well as spindle reciprocation.

Understandably, some coordinated sinusoid motions may not be possible with the mechanical capability of some lathes. Technically these mechanical motions are all feasible if one has the motivation. Notably the Lindow Rose Engine lathe used to create the figures in this article does have a reasonable selection of both pumping and rocking rosettes along with amplitude adjustment but there is no mechanical reciprocation. However, with Linux CNC driver stepper motors on the spindle and two linear axes, there are virtually no constraints on sine wave parameters. Thusly the CNC is a tool that enables accurate exploration of sine wave placement without needing to work around constraints such as rosette bump count, amplitude, or phasing precision. Nevertheless, with a bit of creativity, individuals should be able to reproduce most, if not all, the patterns illustrated here and likely create other interesting variations. Clearly elaborate electronics and controls are not a prerequisite.

Approach:

At the onset of developing sine wave pattern schemes and having seen a demonstration, I sought to create basket weave patterns on a cylindrical surface or even a near cylinder surface modified with a curvilinear apparatus. One way to achieve this is to set a universal cutting frame at a 45-degree angle to the spindle centerline with coordinated sinusoidal motion along the cylinder surface. With each alternate pass, the start of the sinusoidal reciprocating motion is phased 180 rosette pattern degrees from the prior pass and begun in the same physical axial position. Upon indexing a predetermined even number of peripheral positions, the pattern is complete and ever so importantly properly interlaces with the starting pass.

The missing capability for me was the means to generate sine wave motion. Since I already had CNC control for the spindle and two linear axes, a simple path was to create suitable G-code. Since Linux CNC allows named variables, basic math, trigonometric functions, and branching logic, this could be useful for many projects by merely entering appropriate values for diameters, lengths, and pattern characteristics. Once the basket weave pattern was functional, this morphed into additional patterns as I examined other’s work including some Guilloche. This then became an exercise in defining logic and quantifying the positions (degrees and/or inches) of successive waves. Eventually this encompassed four basic motions: two variations of reciprocation, rocking, and pumping.

Sine Wave Basics:

Sine function

A sine wave is a mathematical function in trigonometry. The function for an angle is the ratio of two sides of a right triangle; the side opposite the angle to the hypotenuse (the longest side). The ratio values always fall within the range of minus 1 to plus 1 and the sine of zero degrees is zero. When considered over the range of a full 360 degrees, the value goes from zero to plus one to zero to minus one and back to zero. If multiplied by some value, that value is half the total amplitude. One full cycle, 360 degrees, is one wave length. Individual wavelength times number of waves is total length of a pattern sequence.

Amplitude and Wavelength for Rocking and Reciprocation

Rocking results in the amplitude being on imaginary radial lines from the center of the face. Hence wavelength is degrees around a circle of a chosen radius. The circle is the midpoint of the total sine amplitude.

Reciprocation is wavelength along the radius centerline of the wave form and amplitude is degrees on an arc path from the centerline. If there are 20 radial wave lines, each individual wave is a total of 18 degrees or the amplitude multiplier is 9 degrees. Since the actual amplitude increases with increasing radius, the wave form is altered. As will become evident later in this article, this may or may not be desirable in terms of a pleasing pattern.

Pattern Description:

The patterns are arbitrarily numbered. Some replicate manual work and others are based on a guess about a pattern I may have seen or heard described. In particular, pattern 2 is tangent sine waves from the process described in a prior Rose Engine News article which is available on the Ornamental Turners International website. Because that article has numerous illustrations on the effects of various parameters, the illustrations here will only show the distinction between rocking and reciprocating on the face.

Certain geometries and motions have constraints. The most common is when the wave position change between the last pattern and the first pattern must be identical to the steps between all patterns. This is a necessity for phase shifts along the face radius as the total of all phase shifts must be a multiple of the wavelength. For the rocking mode, this is not a constraint because the last pattern is the outer boundary.

Since the illustrations are done with a pen chuck, an engraving style cutter will provide results similar to the figures when positioned normal to the workpiece face. Other drill type cutter profiles could yield interesting results as would a horizontal cutting platform. There are also a multitude of other design possibilities such as running a rosette as the sine pattern is generated, deliberately omitting lines, or even cutting one pattern over another.

The following paragraphs define each of the nine patterns:

Pattern 1 (Basket weave)

Each successive sine wave begins with an amplitude determined by a sine function which is 180 degrees from the previous wave series. The normal amplitude is the full space of the workpiece radius or full rotation divided by the number of waves.

Pattern 2 (Tangent Series to Series Contact)

This pattern is of sine waves which contact at common slope points. This is achieved by increasing the amplitude by an amount greater than the normally divided space and phasing the start of the sine wave by a calculated amount. For the reciprocation motion, the calculated amount needs to result in the total adjustment being an integer multiplier (K) of 360 degrees for the last pattern in the series to properly interface with the first. This limits the possibilities for reciprocation on both the face or cylinder geometries. For the rocking case, the K value can be anything but must be used with the correct amplitude value.

Pattern 3 (Basket weave with Sublines-Sine Proportions)

This pattern is the same as pattern 1 except that sublines are added between the major boundaries. These sublines are spaced linearly; if three divisions then each division is one third of the table amplitude and phased such that sine motions tend to cut over prior cuts. If the major border lines are cut deeper, this obscures part or all of the overcuts for the minor division cuts.

Pattern 4 (Basket weave with Sublines-Linear Proportions)

Same as pattern 3 except the division spacing and phasing is based on the sine function. For three divisions this is the result of the angle [Arcsine(1/3)] divided by 90 (19.47/90.0 = .216). As will be shown later, the differences between pattern 3 and 4 are subtle.

Pattern 5 (Double Line Basketweave)

Same as pattern 1 except that each two lines are placed some distance apart on each side of the diamond-like patterns between the waves.

Pattern 6 (Barley Corn Like by Double Pass)

This is one sine wave over another with each one phased 180 from the other. This tends to double the number of divisions.

Pattern 7 (Chevron Appearance from Equal Phasing Between Rows and Reverse)

This is a chevron pattern. A fixed number of successive waves are phased some amount plus or minus and then the phasing is reversed for the same fixed number. All phasing is the same value.

Pattern 8 (Moiré Appearance from Sine Function Based Phasing Between Rows and Reverse)

Pattern 8 is similar to pattern 6 except that a total phased amount for phasing is divided by the sine function similar to pattern 4

Pattern 9 (Stacked Waves)

This last pattern is simple stacked waves. In simplest form there is no phasing between cuts. However, phasing can add interest along with amplitudes greater than or less than the normal divided space.

Notes and Observations (Rocking versus Reciprocation)

For comparison purposes and to the extent practicable there is consistency among the patterns generated on the pen chuck. All patterns have an inner diameter of 0.50in, an outer diameter of 3.5in, and 12 features around the center. The geometric space for one twelfth pattern is always 30 degrees and 1.50in radially. Thus all rocking patterns are equivalent to a 12 bump sine rosette and all reciprocating patterns are based on twelve radii. Unless specified otherwise as a variation, there is no added phasing between waves or amplitude greater or less than the geometric space divided by number of features.

In some instances, a high number of reciprocations makes a pattern too busy. In those situations, a smaller number of reciprocations is used and noted in the descriptions and tables. When amplitudes are increased for rocking patterns, pattern ring spacing is adjusted such that the entire pattern remains within the radial boundaries. There is a Figure for each of the nine patterns within the standard geometric boundaries and, in most cases any successive Figure of that pattern involves only one parameter adjustment such as amplitude or some type of phasing pertaining to the start of each wave line. Comparable rocking and reciprocation variations are paired with the odd numbered figures always being rocking and the even numbered always being reciprocation. Changes from the baseline Figure are noted in the tables.

The following paragraphs are some observations about patterns with tables documenting details within the general principles already noted.

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Pattern 1 (Basket weave)

Figures 1 and 2 begin the pattern 1 series with relatively high feature values. The rocking pattern illustrates the sine wave distortion as a function of radius with the pointed nature near the center and a much flatter form at the outermost radius. In contrast the high counts with reciprocation result is very sharp wave forms and the index driven contact points have low line density which creates a depth illusion. The rocking variant could have any number of rings but the outer boundary line would be shifted by 15 degrees for odd values. The reciprocation patterns might be useful as either a background for a strong pattern or a border band. Figures 3 and 4 are similar but have reduced ring and radii values and thus there is more white space.

Figures 5 and 6 have greatly reduced ring and radii values. This increases the sine distortion effects for the rocking cases especially near the center of the figure. For the reciprocation, the nature of the sine distortion effect with radius becomes more apparent than previously. For either, a narrow band near the outer perimeter such as two or three rings or reciprocations would minimize the prominence of this effect.

The remaining figures for Pattern 1 illustrate some simple variations. The first variation is to increase the amplitude by 25% which is shown in Figures 7 and 8. In both cases, the appearance resembles fish net knots at each intersection point. Since borders have not been included and in particular for Figure 8, the ends of each line at both the inner and outer radii are consistent which shows there is no line-to-line phasing.

Table 1 Pattern 1 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
1	1	Rock	12	18
2	1	Recip			18	12
3	1	Rock	12	14
4	1	Recip			14	12
5	1	Rock	12	6
6	1	Recip			6	12
7	1	Rock	12	6			Amp = 125%
8	1	Recip			6	12	Amp = 125%

Rocking	Reciprocation
Figure 1	Figure 2
Figure 3	Figure 4
Figure 5	Figure 6
Figure 7	Figure 8

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Pattern 2 (Tangent Series to Series Contact)

The relationship between amplitude increase and phasing is complex for perfect tangent contact between one line and another. Very well one can determine amplitude and phasing values by trial and error with scrap materials. Once started the changes between wave lines are identical but must be done with some care. A written recipe/checkoff list might be a good way to minimize the likelihood of an error if done manually. Being able to incorporate the mathematics into the G code eliminates the tedium and lessens the likelihood for mistakes once there is validation of the details. Two methods to quantify these movements are explained in footnotes 2 and 4. To visualize results prior to cutting, I created a spreadsheet with plots as small changes can move the contact points and change the entire appearance as can be quickly seen in the referenced newsletter article. Again, the trial-and-error method works but requires patience.

Figures 9 and 10 illustrate the tangent contact effect. For reciprocation there is an obvious helix appearance created from the contacts. In the example this is a left-hand version. Reversing the azimuthal phasing while using the same radial or amplitude direction adjustments will make this helix reverse which is shown in Figures 11 and 12. This effect is less apparent for the reciprocation case but there are a pair of left-hand helixes. Also, notice the different line end points at the perimeter of Figure 10. These reflect the phasing between successive radial lines.

Pattern 2 is an example where lines can be omitted in some repeating manner. When the line density is high, this can add interest. Also, the repeating pattern can be within the repeat such as every 3rd and 5th which results in two lines, a space, a line and repeat. If doing manually a recipe sheet can be helpful along with no interruptions.

Table 2 Pattern 2 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
9	2	Rock	12	6			K = 2
10	2	Recip			6	12	K = 2
11	2	Rock	12	6			Helix = -1, K = 2
12	2	Recip			6	12	Helix = -1, K = 2

Rocking	Reciprocation
Figure 9	Figure 10
Figure 11	Figure 12

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Pattern 3 (Basket weave with Sublines-Sine Proportions)

A unique variation on pattern 1 is to add waves to divide the white space. Figures 13 and 14 illustrate this result. In this example, both the phasing angle and amplitude changes are based on equal division, each relative to the next primary border wave. Because these are sine waves and not sawtooth shapes, the successive waves do not follow the prior lines exactly. Some may view this effect as a positive or a negative feature. To eliminate or reduce, the main borders could be cut deeper than the diving lines and thus hide the effect.

The reciprocation Figure has an irregular edge. Adding a simple circle border diminishes that effect. Also, the line ending can be changed by adjusting the reciprocation by a fraction or by changing the phase starting position at the inner radius. Notably, the reciprocation division lines appear less aligned with the radii for the parameters used in these examples.

Table 3 Pattern 3 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
13	3	Rock	12	6			Subspaces = 3
14	3	Recip			6	12	Subspaces = 3

Rocking	Reciprocation
Figure 13	Figure 14

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Pattern 4 (Basket weave with Sublines-Linear Proportions)

Like pattern 3, pattern 4 is also a variation of pattern 1. The difference between pattern 3 and pattern 4 is the method for spacing the dividing lines. Instead of using a linear division, Pattern 4 division is the sine function. The differences are subtle since the spacing is slightly altered and the overlapping line form changes a bit. As before, cutting the boundary wave a bit deeper can reduce the latter effect. Increasing the number of divisions would make the distinctions more apparent but likely would require increasing the size of the spaces.

Table 4 Pattern 4 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
15	4	Rock	12	6			Subspaces = 3
16	4	Recip			6	12	Subspaces = 3

Rocking	Reciprocation
Figure 15	Figure 16

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Pattern 5 (Double Line Basketweave)

Pattern 5 is also a variation of pattern 1. Instead of a single line, a double line is used. The spacing in these examples is equal to half the total sine wave amplitude or equal to the sine multiplier value. In the first variation, Figures 17 and 18, the lines touch similarly to Pattern 1. For a variation, the patterns are shown with a separation equal to the multiplier amplitude. The consequence of these spacings for set parameters within constant geometric boundaries is that sine amplitudes must be reduced for the added spaces. Comparing Figures 1 and 2 with Figures 17, 18, 19, and 20 illustrate this aspect. If doing manually, the turner needs to do a bit of algebra to determine appropriate values if trying to fill a defined space.

Both the rocking and reciprocation variations have a three-dimensional illusion. This effect would appear to be more evident for the reciprocation variation because of the continually varying line spacing from radius changes. This seems to diminish when the wave sets are separated.

Table 5 Pattern 5 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
17	5	Rock	12	6			Line Space = 1.0 amp
18	5	Recip			6	12	Line Space = 1.0 amp
19	5	Rock	12	6			Line Space = 1.0amp, Pattern Space = 1.0amp
20	5	Recip			6	12	Line Space = 1.0amp, Pattern Space = 1.0amp

Rocking	Reciprocation
Figure 17	Figure 18
Figure 19	Figure 20

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Pattern 6 (Barley Corn Like by Double Pass)

Pattern 6 is Pattern 1 repeated with the wave starts phased 180 pattern degrees. The result shown in Figures 21 and 22 are that line density increases with line crosses and doubling of contact points. Another variant not shown here might be to double or triple one of the passes on each ring or radii.

Table 6 Pattern 6 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
21	6	Rock	12	6
22	6	Recip			6	12

Rocking	Reciprocation
Figure 21	Figure 22

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Pattern 7 (Chevron Appearance from Equal Phasing Between Rows and Reverse)

This pattern is a set number of equal phase adjustments between waves followed by a reversal of those changes for the next set number of waves. As listed in Table 7, there are six adjustments each way. For Figures 23 and 24, each adjustment is 40% of the wavelength and for the second,7/12 of a wavelength. The first adjustments have the left-hand helix which then reverses on the figures. These reversals are on a horizontal line through Figures 24 and 26 and occur mid radius on Figures 23 and 25. Some trial and error with the phasing magnitude could possibly make the effect more apparent as could other adjustments. Likely increasing the number of rocking rings might help.

Table 7 Pattern 7 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
23	7	Rock	12	14			Group = 6, Phase = .4WL/Ring
24	7	Recip			14	12	Group = 6, Phase = 4WL/Radii
25	7	Rock	12	14			Group = 6, Phase = 7/12WL/Ring
26	7	Recip			6	12	Group = 6, Phase = 7/12 WL/Radii

Rocking	Reciprocation
Figure 23	Figure 24
Figure 25	Figure 26

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Pattern 8 (Moiré Appearance from Sine Function Based Phasing Between Rows and Reverse)

As before with Patterns 3 and 4, there are several bases to adjust positions between waves. Similarly, Pattern 8 is the same as Pattern 7 except that the sine function is used for phasing values. The total phased value for the groups is identical but the individual changes within the group differ. Thus, the contact point generating the helixes change and the reversals for the rocking pattern are more evident. All the patterns are within the constant geometric space. If the amplitude were increased slightly, there could be more continuity in the helix line from contacts. As before, the pattern of the individual phasing is most evident for the line end point on the reciprocation variations.

Table 8 Pattern 8 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
27	8	Rock	12	14			Group = 6, Phase = .4WL/Ring (average)
28	8	Recip			6	12	Group = 6, Phase = .4WL/Radii (average)
29	8	Rock	12	14			Group = 6, Phase = 7/12 WL/ring (average)
30	8	Recip			6	12	Group = 6, Phase = 7/12WL/Radii (average)

Rocking	Reciprocation
Figure 27	Figure 28
Figure 29	Figure 30

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Pattern 9 (Stacked Waves)

Of the patterns explored in this article, the simplest to achieve is to merely adjust one position between each sine wave. Figures 31 and 32 illustrate the result for similar patterns used in other examples. These two patterns are within the geometric boundaries of the pattern parameters used for many of the prior patterns. Figures 33 and 34 increase the amplitudes by 25% and the lines encroach those imaginary boundaries. The result is a slight three-dimensional illusion on the flat surface. A different version of this occurs when each wave is phased from the previous as shown in Figures 35 and 36. In this example each increment is 30 degrees of a rosette bump on 360 for the 12 bumps. The next variation combines both the amplitude increase with the wave to wave phasing. This creates the usual helix from the near contacts. Notably, this scheme is a variation on Pattern 2 and could replicate the patterns with specific wave to wave movements. Like pattern 2, the near contact helix can be reversed by changing the phasing directions. Or as with patterns 3 and 4, the chevron or moiré effect can be created. Finally, a 180 degree phasing between successive patterns creates pattern 1.

The final variation is two changes to the baseline figure; increasing amplitude by 50% and decreasing the number of rings and reciprocations. Curiously this appears to create small secondary patterns in the rocking case but not in the reciprocation case as shown in Figures 39 and 40. In both figures white space is increased relative to the prior examples. Using a universal cutting platform on the reciprocation might produce interesting surface contours compared to the engraving cutter.

Table 9 Pattern 9 Parameters

Figure	Pattern	Mode	Rock	Rings	Recip	Radii	Other
31	9	Rock	12	6
32	9	Recip			6	12
33	9	Rock	12	6			Amp = 125%
34	9	Recip			6	12	Amp = 125%
35	9	Rock	12	6			Phase = 360 Deg Total
36	9	Recip			6	12	Phase = 360 Deg Total
37	9	Rock	12	6			Amp = 125%, Phase = 360 Deg Total
38	9	Recip			6	12	Amp = 125%, Phase = 360 Deg Total
39	9	Rock	12	3			Amp = 150%, Phase = 360 Deg Total
40	9	Recip			3	12	Ampl = 150%, Phase= 360 Deg Total

Rocking	Reciprocation
Figure 31	Figure 32
Figure 33	Figure 34
Figure 35	Figure 36
Figure 37	Figure 38
Figure 39	Figure 40

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Conclusions

Regardless of whether motion is based on reciprocation or rocking, some study and comparison is required to grasp setting impacts. Based on the work here, the impacts on reciprocation patterns are less obvious than for the rocking patterns. Seemingly in practice, a larger share of the rocking patterns would appear to be useful than the reciprocation-based patterns.

From experience, pumping and reciprocation-based patterns on a cylindrical surface appear closest to the rocking mode on the face absent the changing wavelength because of changing base circle radius. With parameter adjustment, either motion mode on a cylinder can generate nearly identical patterns just that one is turned 90 degrees from the other.

Lathe accessories and hardware could limit whether or not some of the patterns can be reproduced but with some innovation, many of the patterns and variations should be reproducible. The difficult aspect is likely reciprocation for face cuts.