Rotate Forms About the Axes or Origin

You will learn how to rotate forms about the axes or Origin Point at {0, 0} in the Cartesian Plane. This is a very useful tool to add to your portfolio of tools and is accomplished by multiplying the original {X, Y} coordinates by cos(t)*X + sin(t)*Y and -sin(t)*X + cos(t)*Y to create the new {x, y} coordinates. However, there may be some interim steps involved, especially if Fill Series was used to create a form.

Steps

  • Become familiar with the basic image to create:

The Tutorial

  1. Start by opening a new workbook in Excel from the desktop, from the dock, or from within your Applications folder inside the Microsoft folder. Double click on Excel (either the green X on the dock or the app title in the folder) and select File New Workbook.
  2. Set Preferences. Set R1C1 to unchecked or Off, set Ribbon to checked or On and set Show Formula Bar to checked or On.
  3. Click in the far upper left top corner above the 1 of row 1 and to the left of column A. Doing so will select the entire worksheet. Format Cells Number Number to decimal places 2, show comma. Format Cells Alignment Center. # Title the first worksheet, "Rotate Forms" and save the workbook as "Rotate Forms About The Axes".
  4. Enter the Row 1 and other Column Headers:
    • Enter to cell A1, ORIGINAL_X;
    • Enter to cell B1, ORIGINAL_Y;
    • Enter to cell C1, Degrees_t;
    • Enter to cell D1, New x;
    • Enter to cell E1, New y;
    • Enter to cell F1, X_ADJ;
    • Enter to cell G1, Y_ADJ;
    • Enter to cell H1, Orig_Sqr_X;
    • Enter to cell I1, Orig_Sqr_Y;
    • Enter to cell J1, Adj_Sqr_X;
    • Enter to cell K1, Adj_Sqr_Y;
    • Enter to cell L1, NEW SQR X;
    • Enter to cell M1, NEW SQR Y;
    • Enter to cell C6, Degrees_q;
    • Enter to cell C7, 24; (360º/24º = 15 objects)
    • Enter to cell F6, X2_ADJ;
    • Enter to cell F7, 4;
    • Enter to cell F8, (Square Adjustment)
    • Enter to cell G6, Y2_ADJ;
    • Enter to cell G7, -6;
    • Enter to cell F2, 5;
    • Enter to cell F3, (Circle Adjustment)
    • Enter to cell G2, 5;
    • Enter to cell C2, 180. (By offsetting the circles 180º, they'll be set opposite to one another.)
  5. Define Variable Names:
    • Select columns A,B,D,E,H,I,J and K and do Insert Names Create in Top Row, OK.
    • Select cell range C1:C2 and Insert Name Create in Top Row, OK;
    • Select cell range C6:C7 and Insert Name Create in Top Row, OK;
    • Select cell range F1:G2 and Insert Name Create in Top Row, OK;
    • Select cell range F6:G7 and Insert Name Create in Top Row, OK.
  6. Enter Column Formulas:
    • Enter to cell A2 the formula, =COS((ROW()-2)*PI()/180)+X_ADJ
    • Enter to cell B2 the formula, =SIN((ROW()-2)*PI()/180)+Y_ADJ
    • Select cell range A2:B362 and Edit Fill Down;
    • Enter to cell D2 the formula,
      =COS(Degrees_t*PI()/180)*ORIGINAL_X+SIN(Degrees_t*PI()/180)*ORIGINAL_Y
    • Enter to cell E2 the formula,
      =-SIN(Degrees_t*PI()/180)*ORIGINAL_X+COS(Degrees_t*PI()/180)*ORIGINAL_Y
    • Select cell range D2:E362 and Edit Fill Down;
    • Enter to cell H2 the value, 0, and select cell range H2:H91 and Edit Fill Down. This is the square's X sides being created. Select cell H181 and enter 2. Select cell range H91:H181 and Edit Fill Series Trend, OK. Select cell H181 and copy it to cell range H181:H271. Select cell H361 and enter 0. Select cell range H271:H361 and do Edit Fill Series Trend, OK.
    • Enter to cell I2 the value, 0, and enter to cell I91 the value, 2. Select cell range I2:I91 and Edit Fill Series Trend, OK. This is the square's Y sides being created. Select cell range I91:I181 and Edit Fill Down. Select cell I271 and enter 0. Select cell range I181:I271 and do Edit Fill Series Trend, OK. Select cell I361 and enter 0. Select cell range I271:I361 and do Edit Fill Down, OK.
    • Enter to cell J2 the formula, =H2+X2_ADJ
    • Enter to cell K2 the formula, =I2+Y2_ADJ
    • Enter to cell L2 the formula, =COS(Degrees_q*PI()/180)*Adj_Sqr_X+SIN(Degrees_q*PI()/180)*Adj_Sqr_Y
    • Enter to cell M2 the formula, =-SIN(Degrees_q*PI()/180)*Adj_Sqr_X+COS(Degrees_q*PI()/180)*Adj_Sqr_Y
    • Edit Go To J2:M362 and do Edit Fill Down.
  7. Create the Chart Objects (4):
    • You should end up with 4 series per your worksheet titled 'Rotate Forms':
      • =SERIES(" ORIGINAL",'Rotate Forms'!$A$2:$A$362,'Rotate Forms'!$B$2:$B$362,1)
      • =SERIES(" New",'Rotate Forms'!$D$2:$D$362,'Rotate Forms'!$E$2:$E$362,2)
      • =SERIES(" Adj Square",'Rotate Forms'!$J$2:$J$362,'Rotate Forms'!$K$2:$K$362,3)
      • =SERIES(" NEW SQUARE",'Rotate Forms'!$L$2:$L$362,'Rotate Forms'!$M$2:$M$362,4)
    • Select cell range A2:B362 and do Insert Chart or Chart Wizard or use the Ribbon - Charts, All, Scatter, Smooth Lined Scatter. Add series D2:E362, J2:K362 and L2:M362 -- NOTE! These may not come out right and may require editing in the formula bar!! Extra series may appear, which must be deleted!! You've been warned.
    • Edit in the formula bar the series titles in quotes for each series at the front of the series formula.
    • Pull the Chart Box with the handle that appears when the cursor becomes a double headed arrow over the bottom right corner of the chart, to re-size it so that the circles are nicely true circles, etc. The example Chart Box has the dimensions Width 8.45 by Height 7.86. Herewith is an image of the final chart (try matching the color of your spreadsheet fonts to the color of the data series for each object).

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 Microsoft Excel Imagery, Mathematics, Spreadsheets or 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

  • The squares were made to touch by increasing the line weight of each from 3.75 to 6 pt.
  • Circles: Increase their size and line weights to make them touch in a circular ring.
  • Note that the new circle was rotated 180 degrees from the original circle, not 180 degrees according to degrees of the Unit Circle, and that the New Square was rotated 24 degrees from the Adjusted Square, regardless of degrees according to the Unit Circle. To obtain that information, calculate the degrees of your original object with respect to the Unit Circle and add or subtract thereto/therefrom the new offset degrees (since offset degrees may also be negative).

References

http://mathforum.org/library/drmath/view/54483.html

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