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Converting an Ellipse to a Polygon

Tue, 2006-02-28 17:49 — EditorHuc

Question

How do I draw a polygon with segments of equal length that follow the shape of an ellipse?

Requirements

PowerCADD, any version
PowerCADD version 7.0.1 and Mac OS? X Tiger 10.4.5 or newer recommended


Background

There isn't any single, one step, method or tool to create the geometry you're describing. However PowerCADD does provide various simple techniques that can be combined to create the geometry you're describing.

What To Do

Here's one way to create the geometry you're describing that looks like this:

We've elected to draw the ellipse first, as shown in the screen shot and as illustrated in stage one below for general reference. However, it's important to note you do not need to draw the ellipse first! You could simply draw the multi-sided polygon and use the Edit Window to specify your desired Height and Width values (they don't have to be equal), then proceed directly to stage two. That saves you some steps and is the more likely procedure to be used during production. QTIcon Click Here to see a Demonstration Movie (2.6MB Download) of the quick way to see just how easy it is! (need movie help?), Read on to see all the detailed, step by step instructions.

To create an elliptical polygon with segments of equal length, do this:

Stage One: Draw a multi-sided polygon that matches the shape of an ellipse

  1. Draw an ellipse with the general desired proportions
  2. Option click on the Polygon, Regular tool and specify 12 for the number of sides and click the by vertex icon. You could enter any number of sides you like but it's important to select the by vertex radio button since we're drawing one perfect quadrant of the polygon.
  3. use Center Snap to start drawing the polygon from the center of the ellipse.
  4. use Vertex Snap or End Snap to complete drawing the polygon along the Y axis. In effect the height of the new polygon matches the height of the ellipse
  5. drag the mid point control handle bar and snap it to the right most end point of the ellipse

    6. repeat step 5 for the left hand mid point. The result is a polygon that now matches the width and height of the ellipse. Notice the polygon segments are not of equal length

Stage Two: Set the polygon segments to the desired length

  1. select the polygon and choose Tool menu > Polygon > Distribute Points
  2. Use the Knife tool to break the polygon so you're only working on one quadrant of the ellipse.
  3. select the polygon and choose Edit menu and the Poly Edit... dialog will open
  4. Delete the extra point (point 5 in our example) by pressing Delete
  5. Click the AngleRadio button. Observe the length of each segment of the polygon is displayed
  6. Enter a value in the Length field and press the >> button to advance to the next segment
  7. Enter the same value for this segments length
  8. Repeat for each segment and press Ok
  9. Observe: the result as shown in the earlier screen shot.

Wow! that sure is a lot of steps and it's probably pretty hard to follow. Not to worry, QTIcon Click Here to see a Demonstration Movie (2.7MB Download) of those steps and you'll realize just how fast and easy it is to do! (need movie help?)

In our example we added a few objects such as center lines to make it clearer and we've worked in millimetres to 12 decimal points of accuracy to illustrate how accurate this technique can be. We've also dimensioned the segments and used Edit menu > Mirror to complete the entire shape which could then be Attached to create one contiguous polygon.

Clearly the dimensions we selected for the segment lengths are arbitrary and you may need to be more precise in selecting your segment lengths to suite your particular drawing circumstances.

We hope that helps
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Comments

Mon, 2006-03-06 08:58 — EditorHuc

The Definitive Method

PowerCADD Canada was kind enough to point out that the definitive method is simple and is a combination of some of the methods we've described here.

In this method, the segments of the polygon are all of equal length and the vertices of the polygon touch/match the perimeter of the original ellipse. Again, the method will work with any version of PowerCADD. Here's what the result looks like:

To draw a polygon whose segments are of equal length which touch the profile of an ellipse, do this:

  1. Draw an ellipse, any size
  2. Use the Point tool to insert a Point into the drawing (any location)
  3. Select the point in (2) and choose Edit menu > Cut
  4. Select the ellipse drawn in (1) and choose Edit menu > Objects on Path...
  5. In the Objects On Path dialog, click the Clipboard? radio button, click the Distribute radio button and enter the desired number of segments in the Objects field. For example, enter 12 to create a 12 sided polygon. Press OK
  6. Result: the point from step (3) is inserted 12 times, equally spaced, around the perimeter of the ellipse
  7. use the Polygon tool to connect the points created in step (6).
  8. Result: you have a polygon that connects the points touching the perimeter of the ellipse but the segments are not of equal length
  9. Select the polygon drawin in step (7) and choose Tool menu > Area to insert the perimeter of the polygon into the drawing
  10. Select the numeric value of the perimeter inserted in step (9) and choose Edit menu > Copy
  11. Draw a Line and press Tab to activate the edit window. Tab into the Length field and choose Edit menu > Paste. At the end of the data pasted, type the / symbol (divided by). Enter the number of sides you of the polygon. In our example, we'd enter 12. Press Enter
  12. Result: the line drawn in step (11) is now exactly 1/12 the length of the perimeter of the polgon. Press Tab to activate the Edit Window and Copy the Length of the Line
  13. Select the Polygon drawn in step (7) and choose Edit menu > Edit... to open the Polgon EditDialog
  14. Click the Angle radio button and observe the Len: field (Length) is activated.
  15. Select the contents of the Len: field in step (14) and choose Edit menu > Paste. Click the >> button to proceed to the next point and, again, select the contents of the Len: field and choose Paste. Repeat this step until you've pasted the correct length for every segment
  16. Important you must do step (15) twice. That is, Paste the correct segment length into the Len: field for every segment Twice. This ensures that there aren't any incorrect segment lengths. Press Ok
  17. Result: the polgon now has segments of equal length that touch the perimeter of the ellipse.

To confirm the result, you can select the final polgon and choose Tool menu > Area to insert the Perimeter of the edited polgon.

This may seem like a lot of steps but again, it is much faster to do than it is to read and write. QTIcon Click Here to see a Demonstration Movie (5.2MB Download) of the above stages and steps (need movie help?).

In the demonstration movie we've created a 12 sided polgon. To illustrate the degree of accuracy we've worked to five decimal points in millimetres (that 1/1000 of a millimetre accuracy). In our final result we've dimensioned polygon segments to five decimal points of accuracy in millimetres and you'll note the dimensiosn all match. You will also observer that our original polgon had a perimeter of 118.28942 millimetres. The final, edited, polygon has a peirmeter of 118.28939 millimetres. That a difference of 0.00003 mm or 3/1000th of a millimeter difference in the two perimeters.

We're not familar with any building contractor, site contractor, or other manufacturer that can create a product to 3/1000 of a MILLIMETRE accuracy so it seems that for all practical purposes, the above technique is accurate for real world purposes cool

Wed, 2006-03-01 09:26 — EditorHuc

One more quick and dirty method

here's one more quick and dirty method to make a polygon that matches the profile of an ellipse

in this example we'll just make a 10 sided polygon that fits the X and Y (height and width) of an existing ellipse in the drawing.

  1. choose the Ellipse tool and draw an Ellipse
  2. press option and click on the Polygon, Regular tool
  3. special 10 sides and click the by center to vertex radio button
  4. use Center snap to draw the new polygon from the Center fo the ellipse
  5. use Vertex snap to end the drawing operation for the polygon at the top most point of th ellipse
  6. stretch the right most mid point control point of the polygon to match the right most point of the ellipse
  7. repeat step (6) for the left side of the polygon

QTIcon Click Here to see a Demonstration Movie (1MB Download) of the above stages and steps (need movie help?)

Wed, 2006-03-01 09:15 — EditorHuc

Another Method...

if you don't need the segments to be of equal length, there's another, very fast, option

To draw the segments of a polygon that fits the profile of an ellipse, whose segments are different lengths, do this:

  1. choose the elliptical arc tool and draw one quadrant of the ellipse
  2. choose the Point Tool and click to insert a single point anywhere in the drawing
  3. select the point in (2) and choose Edit menu > Cut
  4. select the arc in (1) and choose Tool menu > Objects On Path...
  5. in the objects on path dialog box, click the Clipboard? radio button, click the Distribute radio button and enter the desired number of items in the objects field. For example, if we want to draw 3 segments, insert 4 objects on the path
  6. Press OK and the points are inserted on the path of the ellipse
  7. choose the Line tool and draw lines to connect the dots
  8. now simply select the line segments, and mirror them to create the rest of the shape

QTIcon Click Here to see a Demonstration Movie (2.0MB Download) of the above stages and steps (need movie help?)

Wed, 2006-03-01 08:28 — EditorHuc

To simply convert an ellipse to a polygon

I should have also pointed out that if all you need is an ellipse simply converted to a polygon, just:

  1. Select the Ellipse
  2. choose Tool menu > Convert to Polygon
  3. the ellipse is now a polygon

But be careful, an ellipse is what we refer to as a piece of high level geometry. That is, once you make a polygon out of it, you can't convert it back to an ellipse.

However, once the ellipse is converted (and there are multiple levels of conversion accuracy available when making the polygon), you can use the various options available in the Tool menu > Polygon submenu to further modify the geometry.

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