Ellipses and tools to construct them

You access the ellipses-tools by pressing the drop arrow:

[ellipses_menu]

To read here you should be familiar with Generalities on tools

[ellipse-characteristics]

[bullet] Ellipse characterisitics

Closed curve of second degree, implemented as a set of points (initially 60) joined by segments. The following four construction methods make it appear always in a scheme-component. Has no anchor points but is modifiable through its masters. To draw quickly an ellipse use the foci+pt tool. For finer drawings, the Add points tool augments the interpolation points. Corresponding symbolic constant = kEllipse. The menu Conics General contains a variety of tools defining conics, which some times are ellipses. For example through the button   [through5]   you get the tool laying a conic through 5 points. For appropriate places of the 5 points the resulting conic is an ellipse.

[bullet] Ellipse from foci and point on it

Menu-item: Ellipse \ Ellipse(2 foci + pt)
Keyboard shortcut:   None
Description
Operates in three constructive stages:
In stage1(X,A), you set the first focal point at A. In stage2(Y,B) you set the second focal point at B. In stage3(Z,C) you set a point C through which the ellipse should pass. The tool calculates the unique ellipse satisfying these conditions and creates an object of type kEllipse and a scheme-component of type kEllipseFPt.       Picture
Parameters
None
Remark
The points A, B, C are masters and changing them modifies the ellipse.

[bullet] Ellipse from its axes a, b

Menu-item: Ellipse \ Ellipse(a, b) _
Keyboard shortcut:   None
Description
Operates in two stages:
Stage1(A,B) is selective and selects at A a segment/line/side x of an object o1, whose direction will be one of the axes of the ellipse. Stage2(C,D) is constructive and defines the axes a, b using the coordinates of D and the segment x. The tool creates the ellipse enclosed in the rectangle whose middle point is the middle of x, one of its vertices is D, and one of its sides is parallel to x.

[ellipse-by its axes]

More precisely, the tool creates a point object o2 at D and a group of three objects: the ellipse and its two focal points, depending on o1 and o2. The scheme component created is of type kEllipseAB.
Parameters
One integer identifying the side x of o1.
Remark
The ellipse is modifiable through o1 and o2(D). The dotted lines are not drawn.

[bullet] Ellipse from foci and major axis

Menu-item: Ellipse \ Ellipse(2a + 2foci) _
Keyboard shortcut:   None
Description
Operates in three stages:
In stage1(X,A), you select a segment/side x of an object o1, defining the length 2a of the major axis of the ellipse. In stage2(X,B), stage3(Y,C) you set the focal points at B and C. If the data are compatible, then an ellipse is created depending on o1, B and C. The tool creates an object of type kEllipse and a scheme-component of type kEllipseABL.       Picture
Parameters
One integer determining the side x of o1, from which 2a=|x| is calculated.
Remark
The compatibility condition for 2a and the two foci B, C is that the distance |BC|<2a. Then and only then there is an ellipse with these data. Every point P on this ellipse satisfies |PB|+|PC|=2a.

[bullet] Ellipse from Directrix one focus and a pnt on it

Menu-item: Ellipse \ Elli/Hype(Directrix+focus+pt) _
Keyboard shortcut:   None
Description
Operates in three stages:
In stage1(X,A), you select a segment/side/line x of an object o1, defining the directrix of the ellipse. In stage2(X,B) you set one focus point at B and in stage3(Y,C) you set a point C through which the ellipse/hyperbola has to pass. Depending on the relative position of these three data the tool constructs either an kEllipse or an kHyperbola and a scheme-component of type kConicDFPt.       Picture
Parameters
One integer determining the side x of o1 whose supporting line is taken as directrix.
Remarks
a) The focal point B, defined in stage2, is considered to be the polar-point of the directrix w.r. to the under construction conic.
b) The point C, supposed to be on the conic, determines the eccentricity of the conic: e = |CB|/distance(C,x). If e<1 we get an ellipse. If e>1 an hyperbola. The case e=1 corresponds to a parabola, fully determined by x and B.

When Newton saw an apple fall, he found
    In that slight startle from his contemplation-
'Tis said (for I'll not answer above ground
    For any sage's creed or calculation)-
A mode of proving that the earth turn'd round
    In a most natural whirl, called 'gravitation';
And this is the sole mortal who could grapple,
Since Adam, with a fall, or with an apple.
    Byron, Don Juan, Canto X, 1