# How to Create Spheroidal Asymptotes and Skewed Sphere Ring

Four Parts:The TutorialExplanatory Charts, Diagrams, PhotosOptional Charts, Diagrams, PhotosHelpful Guidance

It is unusual in math, usually, to find patterned asymptotes. In this article, you'll learn to make spheroidal asymptotes as pictured below and a skewed sphere-ring.

## Steps

### Part 1 The Tutorial

- 1
**Start by opening a new Microsoft Excel workbook and creating 3 worksheets:**DATA, CHART (unless you use Chart Wizard) and SAVES. Save the workbook as "Spheroidal Asymptotes and Skewed Ring", or as you think best. A related file has been previously saved as "THE Garthwaite Curve" since many past problems were solved by it. - 2
**Set Preferences:**Open Preferences. Recommended Settings: Set General to R1C1 Off and Show the 10 Most Recent Documents; Edit - set all the Top options to checked except Automatically Convert Date System. Display number of decimal places = blank (for integers preferred), Preserve display of dates and set 30 for 21st century cutoff; View - show Formula Bar and status bar, hover for comments and all of Objects, Show gridlines and all boxes below that auto or checked; Chart - show chart names and data markers on hover. Leave rest unchecked for now; Calculation -- Automatically and calculate before save, max change .000,000,000,000,01 w/o commas if you do goal-seeking a lot and save external link values and use 1904 system; Error checking - check all; Save - save preview picture with new files and Save AutoRecover after 5 minutes; Ribbon -- all checked except Hide group titles and Developer. - 3
**Input the Defined Variable Names in Row 1.**A1: AjRows; B1: GM; C1: Factor1; D1: KEY; E1 Number; F1 Format Cells Aligned Left enter "968115/(25680*2)=18.84959112 which divided by π = 6.000011204" w/o quotes. Select cell range A1:E2 and Format Cells Alignment Center. - 4
**Input the Variables in Row 2.**A2: 2880; B2 "=(-(1-SQRT(5))/2)^IF(Spheroids<24,1,1)" w/o quotes; C2: .125; D2: w/o quotes "=IF(Spheroids<=24,Spheroids*VLOOKUP(Spheroids,LOOKER2,2)*2,Spheroids)"; E2: 1; F2: Cell Format Align Left and enter w/o quotes "=25680*2*6*PI()/15"; G2: Enter w/o quotes "=1712*12*PI()". - 5
**Select cell range A1:**E2 and Insert Name Create Names in Top Row. Select A2:E2 and Format Cells Border bold Black Outline Center (divider). That signifies they're input cells of a sort. - 6
**Format Cells for cell range F2:G2 Number Decimals 4.**In fact, just do this for the entire worksheet by clicking between the A and the 1 in the far upper left corner to select the entire worksheet and Format Cells Number Number Decimal Places 4. Align Center if you like as well. You'll need to go back and align the note left in cell F1 if aligning all center. - 7
**Enter the Variable Defined Names for Row 3.**A3: Tip; B3: Base; C3: Spheroids; D3: ShrinkExpand; E3: PiDivisor; F3: Thickness; G3: ShrinkExpand2; H3: VARIABLE. - 8
**Enter the valriable values or formulas for Row 4:**A4: "=Base*12/(VARIABLE*1)*PI()" w/o quotes; B4: "=16*107" w/o quotes; C4: 12; D4: 1; E4: 180; F4: 1; G4: 1; H4: 1. - 9
**Select cell range A3:**H4 and Insert Name Create Names in Top Row. Select range A4:H4 and Format Cells Border Black bold Outline Center (divider). - 10
**Enter the Column Headings in Row 5.**A5: Base t; B5: constant c; C5: Cos; D5: Sin; E5: Main X; F5: Main Y; G5: Second X; H5: Second Y. - 11
**Select cell L8 and enter LOOKER2.**Select cell L9 and input 1. Edit Go To cell range L9:L32 and do Edit Fill Series Columns Linear Step Value 1 OK. - 12
**Enter the following values into M9:M32.**M9: 24, M10: 16, then continue down as follows: 8, 4, =PI(), =PI(), =PI(), 2 from M16 down to M24, then 1 from M25 to M32. - 13
**Edit Go To cell range L9:M32 and Insert Name Define Name LOOKER2 for cell range $L$9:**$M$32. Format Cells Border Red Bold Outline and Fill yellow. There should no longer be any errors in the Defined Variables Section. If there are any, see ERRORS in the Warnings section below please. - 14
**Enter the column formulas:**- In cell A6, enter the formula w/o quotes "=IF(ODD(Spheroids)=Spheroids,0,Tip)" and do Insert New Comment and edit in "Original Formula =IF(ODD(Spheroids)=Spheroids,0,Tip)". See Tip below concerning Inserting New Comments for all original values and formulas. Format Cell Fill Light Blue to make it distinct from other formulas in the same column, since it will not simply be filled down.
- Edit Go to cell range A7:A2886 and with cell A7 the active high-lighted cell, enter the formula, "=((A6+(-Tip*2)/(AjRows)))" w/o quotes and do Edit Fill Down. Select cell A7 and Insert New Comment and edit in "Original formula =((A6+(-Tip*2)/(AjRows))) down to A2886." That is there in case someone decides to get creative ....
- In cell B6, enter the formula w/o quotes "=IF(Spheroids<=24,Base*24/Spheroids,Base*24/Spheroids)" which I realize is redundant, but it used to solve a problem, so it's being retained.
- Edit Go To cell range B7:B2886 and with B7 active and high-lighted, enter the formula, "=B6" w/o quotes and do Edit Fill Down. Do Insert New Comment and edit in "Original formula =B6 down to B2886 as provides the constant value, per the adjusted value in B6." Expand the comment frame to fit.
- Edit Go To cell range C6:C2886 and with C6 active and high-lighted, enter the formula w/o quotes, "=Thickness*Spheroids/KEY*(COS((ROW()-6)*Number*PI()/PiDivisor*Factor1))" and do Edit Fill Down, then do Insert New Comment and edit in "Original formula =Thickness*Spheroids/KEY*(COS((ROW()-6)*Number*PI()/PiDivisor*Factor1)) down to C2886. Thickness is new and so is the formula for Key -- both have solved some previous problems with partial circles and overlaps (or 'superpositioning'). The user may still build the Factor1 formula and Lookup Table from previous wikiHows if so desired."" Expand the comment frame to fit.
- Edit Go To cell range D6:D2886 and with D6 active and high-lighted, enter the formula w/o quotes, "=Thickness*Spheroids/KEY*(SIN((ROW()-6)*Number*PI()/PiDivisor*Factor1))" and do Edit Fill Down, then do Insert New Comment and edit in "Original formula =Thickness*Spheroids/KEY*(SIN((ROW()-6)*Number*PI()/PiDivisor*Factor1)) down to C2886. Thickness is new and so is the formula for Key -- both have solved some previous problems with partial circles and overlaps (or 'superpositioning'). The user may still build the Factor1 formula and Lookup Table from previous wikiHows if so desired." Expand the comment frame to fit.
- Edit Go To cell range E6:E2886 and with E6 active and high-lighted, enter the formula w/o quotes,"=((SIN(A6/(B6*2))*GM*COS(A6)*GM*(COS(A6/(B6*2)))*GM)+C6)/ShrinkExpand" and do Edit Fill Down. Do Insert New Comment and edit in "Original Formula =((SIN(A6/(B6*2))*GM*COS(A6)*GM*(COS(A6/(B6*2)))*GM)+C6)/ShrinkExpand where GM is the Golden Mean which keeps things proportionate and C6 is the ring's cosine at 0, =1, while ShrinkExpand allows the user to modify the graph's size relative to ShrinkExpand2; otherwise this is the formula for a spherical helix from the CRC manual with the z dimension added into the x dimension." Expand the comment frame to fit.
- Edit Go To cell range F6:F2886 and with F6 active and high-lighted, enter the formula w/o quotes,"=((SIN(A6/(B6*2))*GM*SIN(A6)*GM*(COS(A6/(B6*2)))*GM)+D6)/ShrinkExpand" and do Edit Fill Down. Do Insert New Comment and edit in "Original Formula =((SIN(A6/(B6*2))*GM*SIN(A6)*GM*(COS(A6/(B6*2)))*GM)+D6)/ShrinkExpand where GM is the Golden Mean which keeps things proportionate and D6 is the ring's sine at 0, =0, while ShrinkExpand allows the user to modify the graph's size relative to ShrinkExpand2; otherwise, this is the formula for a spherical helix from the CRC manual with the z dimension added into the y dimension." Expand the comment frame to fit.
- Edit Go To cell range G6:G2886 and with G6 active and high-lighted, enter the formula w/o quotes,"=((SIN(A6/(B6*2))*GM*COS(A6)*GM*(COS(A6/(B6*2)))*GM)+C6)/ShrinkExpand2" and do Edit Fill Down. Do Insert New Comment and edit in "Original Formula =((SIN(A6/(B6*2))*GM*COS(A6)*GM*(COS(A6/(B6*2)))*GM)+C6)/ShrinkExpand2 where ShrinkExpand2 allows the user to modify the graph's size relative to ShrinkExpand for Main X and Y." Expand the comment frame to fit.
- Edit Go To cell range H6:H2886 and with H6 active and high-lighted, enter the formula w/o quotes,"=((SIN(A6/(B6*2))*GM*SIN(A6)*GM*(COS(A6/(B6*2)))*GM)+D6)/ShrinkExpand2" and do Edit Fill Down. Do Insert New Comment and edit in "Original Formula =((SIN(A6/(B6*2))*GM*SIN(A6)*GM*(COS(A6/(B6*2)))*GM)+D6)/ShrinkExpand2 where ShrinkExpand2 allows the user to modify the graph's size relative to ShrinkExpand for Main X and Y." Expand the comment frame to fit.

### Part 2 Explanatory Charts, Diagrams, Photos

- 1
**Create the chart.**Edit Go to cell range E6:F2886 and from the Ribbon or Chart Wizard, select Chart, All, Scatter, Smoothed Line Scatter (use the scroll box -- it's down a ways). If in Chart Wizard, a new Chart Window will automatically open. Otherwise, a chart will appear on the DATA worksheet to be copy or cut and pasted into the upper left corner of the CHART worksheet. Hover the cursor over the lower right hand corner of the chart box area until the double-headed arrow appears, then pull and expand it into a large square. Double-click on the series plot line to bring up its dialog box and select Line Weight 1 and Color Dark Blue, or as you prefer. Set Transparency. Back on the Chart, click in the Plot Area and select Chart Layout from the Ribbon, where you can get rid of gridlines, legend, set or delete axis titles and input a Chart Title or Text Box (with direct editing), do 3-D formatting, Glow, Bevel, Soft Edges, etc.- Click in the Plot Area and hold down the Shift Key and do Edit Copy Picture. Then activate the SAVES worksheet and, still holding down the Shift Key, Paste Picture. The Option Key allows you to paste a Microsoft Office Drawing Object and do other special effects, plus, it remains linked to the data. Change the data now by entering 1.5 in cell F4 (Thickness) and go back and see the effect on the chart. -- more spread out, but a bit less spherical. Go back to DATA and enter 60 for Spheroids in cell C4. Pretty good little spheres. Set F4 to 1 and look again. Too much overlapping. Set F4 to 2. Just right, approximately.
- Click in the Plot Area and do the Chart Add Data menuitem. For Range, respond by going back to the DATA worksheet and selecting cell range G6:H2886. Mine comes out an oval and when I click on the new series plot, the formula bar reads, "=SERIES(,Data!$E$6:$E$2886,Data!$H$6:$H$2886,4)" which is wrong. Edit the E's to become G's. You may get a diagonal line which has a series of "=SERIES(,Data!$E$6:$E$2886,Data!$G$6:$G$2886,3)" not wanted at all, so delete it from the Formula Bar with a backspace and return. Activate the DATA worksheet and enter 1 for Spheroids in cell C4 and then check out the CHART worksheet.Go back to DATA and enter a Thickness of 1 in F4 and then re-check the CHART. Click on the Added Data Series and delete it. Grab the lower right corner of your Chart Area and adjust the frame until there is a nice sphere shape, which happens when it's about 5" wide by 4" tall. With the Shift Key down, do Copy Picture and activate SAVES worksheet and do Paste Picture with the Shift Key depressed. Double click in the Plot and Chart Areas (near the edge) and set them to No Fill (transparent) and No Line. Double-click on the sphere and to the lower right of the Picture Styles box is a Brush in a round circle with a down arrow; select it, and choose Shadow Perspective - the one floating underneath. You should see a round shadow if you have set the background to Transparent. Save the workbook.
- The Asymptotes Formula Input. Return to the DATA worksheet. Enter 64 for Spheroids in cell C4, 2 for Thickness in cell F4 and 1 for Number in cell E2 (each of which has an effect on the outcome).

- 2
**Create a new worksheet in the workbook titled "Asymptotes Chart"**- Edit Go To cell range R6:R2886 and with cell R6 active and high-lighted, enter the formula, "=E6" and Edit Fill Down. Edit Go To cell range S6:S2886 and with cell S6 active and high-lighted, enter the formula, "=F6" and Edit Fill Down. You may Insert New Comments for these; it would be a good idea.
- Edit Go To cell range P6:P2886 and with cell P6 active and high-lighted, enter the formula, "=((R6/(R6-1))*PI()/180)". and Edit Fill Down. Select cell P6 and Insert New Comment, "Original formula =((R6/(R6-1))*PI()/180), which is Neutral Operations; google NeuOps-Graphs01 to learn more".
- Edit Go To cell range Q6:Q2886 and with cell Q6 active and high-lighted, enter the formula, "=((S6/(S6-1))*PI()/180)". and Edit Fill Down. Select cell P6 and Insert New Comment, "Original formula =((S6/(S6-1))*PI()/180), which is Neutral Operations; google NeuOps-Graphs01 to learn more. Basically, if we have two numbers a and b, the operators of Addition and Multiplication will be Neutral versus one another when a=b/(b-1) and b=a/(a-1) for a+b=a*b=c where a^2/(a-1)=b^2/(b-1) = c. This is true for F=ma=m-a, E=mc^2=m+c^2, a^2+b^2=c^2=a^2 * b^2 = a^4/(a^2 -1) =b^4/(b^2 -1), ab = a^b, a+b = a^b via modified Newton-Raphson, etc, etc." Expand comment frame to fit.
- Inspect cell range P6 to Q2886 and delete all ############ overflows of exceptionally large numbers -- just delete them please.

### Part 3 Optional Charts, Diagrams, Photos

- 1
**Create the new Chart**- Edit Go To cell range P6:S2886 and from the Chart Wizard or Ribbon select Chart, All, Scattered, Smoothed Line Scattered. If in Chart Wizard, a new Chart sheet will automatically be created. Otherwise, a new chart will appear on the DATA worksheet to Copy or Cut and Paste to the Asymptotes Chart worksheet's upper left corner. Hover over the lower right corner until there appears the double headed arrow and pull the chart open so that it occupies the window,
- Double-click on the horizontal axis and set the Minimum to -.1 and the Maximum to .1; then double-click on the vertical axis and set the Minimum to -.1 and the Maximum to .1; Double-click on the series plot lines and change their color and weight to suit your tastes. Shift Key down and Copy Picture and go to SAVES worksheet and Shift Key down Paste Picture and make notes of Data Settings.
- Skewed Sphere Ring: Set Spheroids to 12, Thickness to 1.75 and Number to (Hold off). Create a new Chart from the same data with the above changes of Vertical MaxMin ± .6 and Horizontal -.025 and Maximum .01, then Copy Picture with Shift Key and with Shift Key Paste Picture to SAVES worksheet and make notes of Data Settings. Now change Number from 1 to 2. Copy Picture with Shift Key and with Shift Key Paste Picture to SAVES worksheet.
- Now double-click the Vertical Axis and reset to MaxMin of ± .05 to see the other series! Copy Picture and Paste Picture with Shift Key and make note of data settings.

### Part 4 Helpful Guidance

- 1
**Make use of helper articles when proceeding through this tutorial:**- See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
- For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

## Tips

- It's a good idea to Insert New Comment for all the variables and formulas as they were originally input, since you'll probably want to change them and see what the effects are. It's a lot easier to get the formula by Edit Comment than to come back to wikiHow, find this article and track down the original formula or variable value.
- So what changed to solve the problems?
- I quit concentrating on FACTOR1 as the problem and focused instead on KEY. Factor1's formula still reads "=VLOOKUP(ABS(Spheroids),Looker,IF(Spheroids<=24,2,2))" but above I gave it to you as simply .125, because column 2 of the Looker Lookup Table only contains that value, and that is the correct value to make complete rings in the COS and SIN formulas. There are 2880 rows and 2880 rows / 360 = 8. So I needed to take 1/8th of the value each row, or .125 as a Factor to make 1 complete ring with no partial rings (as caused by the old #3 column of the LOOKER lookup table, despite the fact that it worked rather well for keeping sphere-ness to each sphere) or overlapping.
- The KEY formula, as used by COS and SIN as factors OUTSIDE OF their trig functionality (unlike Factor1), was changed to work only for SPHEROID quantities less than 24: "=IF(Spheroids<=24,Spheroids*VLOOKUP(Spheroids,LOOKER2,2)*2,Spheroids)", which for Spheroids<+24 are multiplied by 2 ... or the Table could be multiplied by 2, either way. But understand the COS (and SIN) formulas, "=Thickness*Spheroids/KEY*(COS((ROW()-6)*Number*PI()/PiDivisor*Factor1)" are dividing the Spheroids by KEY. In the case where Spheroids>24, Key=SPHEROIDS, so the COS formula reduces to SPHEROIDS/SPHEROIDS = 1, null effect. In the most extreme case where SPHEROIDS = 1, the LOOKER2 table gives a value of 24, so we have SPEROIDS/KEY = 1/(2*24) = 1/48th, or 0.0208333333333333, applied over 2880 rows = exactly 60 rows of datapoints to a given spiral ring of the single sphere, which is adequate detail. If desired the rest of the calculations can be made in response to any inquiry concerning any value(s) of the LOOKER2 Table. See Step 12.
- For additional corrections, especially for large quantities of SPHEROIDS, there is a new adjustment factor called Thickness. It is actually the entire ring size, so increasing it allows more spheroids and their detail to show without overlapping.
- These changes allowed simplification of the Main X and Main Y formulas, and others like them.

## Warnings

- ERRORS: If you have errors or error values, either the sheet in incomplete and needs the Lookup Tables for critical variables or perhaps you've made a mistake somewhere along the line. If the instructions have been completed and there are still errors, select the cell that has the error value that is furthest left and topmost first. Look for a typo in a formula or unmatched parentheses. Possibly, a Defined Name is wrong -- they need to be input into the formulas exactly as they were defined. Do Insert Name Define to check. If you have DIV/0!, I do not, so look for a variable that somehow did not get filled in with a value perhaps. At any rate, what you want to do is select the cell with the error, and after checking all those typical errors, do Tools Auditing Trace Precedents and/or Trace Error. If fixing a few topmost leftmost errors does not fix the rest of your errors on your worksheet, you may need to do it the hard way, from the bottom right upwards then leftwards.

## Things You'll Need

## Sources and Citations

- "CRC Standard Curves and Surfaces", David von Seggern, 1993, CRC Press Inc., Boca Raton, FL. ISBN 0-8493-0196-3, Library of Congress Card Number 92-33596, [pp. 264 7.1.4 Spherical Helix]
- The source workbook for this article is "THE Garthwaite Curve Asymptotes.xlsx"

## Article Info

Categories: Mathematics | Microsoft Excel