Constructing Ellipses

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Constructing Ellipses

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay First-time Visitors: Please visit Site Map and Disclaimer . Use "Back" to return here.

Foci and Axes

Given the major and

minor axes of an ellipse,

you can always find the

foci. You need the foci for

some construction

methods. Just draw radii

of length a from the endsof the minor axis.

Given the foci, however,

you can't uniquely

determine the axes. You

need additional

information such as the

length of one axis.

However, the major axis

is always along the line

through the foci and the

minor axis always

perpendicularly

bisects the line between

the foci.

Pin and String

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Mostly useful as a

heuristic tool to help

students visualize

ellipses, but it can be

useful for constructing

large ellipses.

Put a pin in each focus

and tie a string to each

pin leaving slack with

length 2a. Pull the string

taut with a pencil point

and slide the pencil to

draw the ellipse.

This makes use of thefact that an ellipse is the

locus of points whose

distances from two fixed

points have a constant

sum.

http://www.sahraid.com

/ has a spiffy tool that

makes this method much

handier to employ.

Trammel Method

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In my view the

best method.

With the major

and minor axes

constructed (andextended) mark a

piece of paper

with points O, A

and B so that OA

= a and AB = b.

Slide O along the

minor axis and B

along the major

axis. Point A

traces out the

ellipse.

With any

reasonable care

this method is

quite accurate

and it is very fast.

For cases where

the axes are

similar in size, the

method above

may be

inaccurate. It is

also possible to

draw an ellipse

using an external

trammel.

Envelope Method



Given the focus and a

circle whose diameter

equals the major axis,

draw a radius from the

focus to the circle, then a

line perpendicular to

that. The perpendiculars

sweep out the ellipse.

This is fast and can be

quite accurate. If you use

an index card to draw the

perpendiculars you can

dispense with drawing

the radii. Just let one

edge of the card serve asa radius and use the

other to draw the

perpendicular.

For a quick and dirty way

to sketch an ellipse this

rivals the trammel

method. It roughs out the

area enclosed by the

ellipse a bit faster.

The Draftsman's Method



This method is given in a

lot of drafting texts. For

extreme accuracy it'sprobably the best

method. It's convenient

for use on a drafting

board with T-square and

triangles.

Construct the major and

minor axes and draw

circles with each axis as

diameter. Also construct

radii as shown. Angles

aren't critical so radii can

be closer in areas where

greater accuracy is

needed.



Draw vertical lines from

the intersection of each

radius with the outer

circle and horizontal lines

from the intersection of

each radius with the

inner circle. The

intersection of each pair

of corresponding lines is

a point on the ellipse.

It's easy to see this is

simply animplementation of the

parametric equation of

an ellipse: x = a cos t, y =

b sin t.



When the desired

number of points are

drawn, construct the

ellipse.

Paralellogram Method



Sometimes you know an

ellipse can be enclosed

within a parallelogram,

for example, a

foreshortened view of a

circle or a spherical

object sheared out of

shape.

Construct the

parallelogram and divide

it into quarters. Divide all

the lines into equal

numbers of segments

and number them as

shown.

From the midpoints of

opposite sides, draw

straight lines through the

tick marks as shown.

Continue the

construction for all

quadrants of the

parallelogram.



Connect all the 1-1, 2-2

intersections, etc., to

construct the ellipse.

This construction will

work perfectly well if theparallelogram is a

rectangle, so it will work

to construct an ellipse if

the major and minor axes

are known.

If the parallelogram is a

square, the resulting

ellipse is a circle. The

intersecting lines areperpendicular, and the

construction is the

famous one of

constructing a right angle

inside a semicircle. The

general construction here

simply works by

deforming the

construction so an ellipse

results.

Five-Center Method



This was long a standard

in drafting texts, and

pretty much obsolete

today. Any CAD program

will draw true ellipses,

and the trammel and

envelope methods are

faster. It's here mostly

out of respect for

tradition.

Draw the major and

minor axes of the ellipse

and draw a rectangle

around them. Construct

XY and draw P-C1perpendicular to XY.

Let X-C2 = r and locate Q

such that YQ = 2r. Draw a

circle of radius r centered

on C2. By symmetry,

everything on the right

side of the diagram is

repeated exactly on the

left.

Using C1 as center and

C1-Q as radius, draw an

arc through Q. Call the

intersection of this arc

with the circle X-C2 C4.



Construct line C1-C4 and

extend it to T. Using C1

as center, draw arc YU.

Construct line C4-C2 and

extend it to L. Using C4 as

a center, construct arc

UV. The successive arcs

YUVX are an

approximation to theellipse.

The true ellipse is shown

in red, the approximation

in purple. The

approximation is quite

good for slightly or

moderately eccentric

ellipses but becomes

obviously incorrect for

very elongated ellipses.



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Created 17 June 2005, Last Update

Not an official UW Green Bay site

B est m atches f or e llipse c onstruction

G iven t he m ajor a nd m inor a xes o f a n e llipse, y ou

c an a lways f ind t he f oci. Y ou n eed t he f oci f or s

ome c onstruction m ethods. J ump t o t ext»

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