Hyperbolas and tools to construct them

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

[hyperbolas_menu]

To read here you should be familiar with Generalities on tools

[hyerbola-characteristics]

[bullet] Hyperbola characterisitics

Open, two components, symmetric curve of second degree, implemented as a set of points (initially 60 for each component) 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 Hyperbola use the foci+pt tool. For finer drawings, the Add points tool augments the interpolation points. Corresponding symbolic constant = kHyperbola. The menu Conics General contains a variety of tools defining conics, which some times are hyperbolas. 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 a hyperbola.

[bullet] Hyperbola from foci and point on it

Menu-item: Hyperbola \ Hyperbola(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 hyperbola should pass. The tool calculates the unique hyperbola satisfying these conditions and creates an object of type kHyperbola and a scheme-component of type kEllipseFPt.       Picture
Parameters
None
Remark
The points A, B, C are masters and changing them modifies the hyperbola.

[bullet] Hyperbola from its axes a, b

Menu-item: Hyperbola \ Hyperbola(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 hyperbola. Stage2(C,D) is constructive and defines the axes a, b using the coordinates of D and the segment x. The tool creates the hyperbola whose axes coincide with x and the orthogonal to x, with center the middle of x and a, b determined by the projection of D on the previous axes.

[hyperbola-by its axes]

More precisely, the tool creates a point object o2 at D and a group of three objects: the hyperbola 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 hyperbola is modifiable through o1 and o2(D). The dotted lines as well as the arrows are not drawn.

[bullet] Hyperbola from foci and major axis

Menu-item: Hyperbola \ Hyperbola(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 hyperbola. In stage2(X,B), stage3(Y,C) you set the focal points at B and C. If the data are compatible, then a hyperbola is created depending on o1, B and C. The tool creates an object of type kHyperbola 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 a hyperbola with these data. Every point P on this hyperbola satisfies ||PB|-|PC||=2a.

[bullet] Hyperbola 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/hyperbola. 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.

[bullet] Rectangular hyperbola by its center and one vertex

Menu-item: Hyperbola \ Rectangular Hyperbola
Keyboard shortcut:   None
Description
Operates in two constructive stages:
In stage1(X,A), you set a point o1, at A, defining the center of the rectangular hyperbola. In stage2(X,B) you set a point o2 at B, defining one vertex of the hyperbola under construction. The tool constructs a kHyprbola and a scheme-component of type kHyperbolaR.       Picture
Parameters
None.

[bullet] Conjugate hyperbola

Menu-item: Hyperbola \ Conjugate Hyperbola _
Keyboard shortcut:   None
Description
Operates in one selective stage:
In stage1(X,A), you select at A a hyperbola. The tool constructs a kHyprbola and a scheme-component of type kHyperbolaConj, which is the conjugate hyperbola of the selected one. The conjugate hyperbola has the same center, axes and asymptotes with the original hyperbola.       Picture
Parameters
None.

[bullet] Asymptotic lines

Menu-item: Hyperbola \ Asymptotic lines _
Keyboard shortcut:   None
Description
Operates in one selective stage:
In stage1(X,A), you select at A a hyperbola. The tool constructs a group of two lines and a scheme-component of type kAsymptotic.       Picture
Parameters
None.

The cause of this effect, or this defect,-
    'For this effect defective comes by cause,'-
Is what I have not leisure to inspect;
    But this I must say in my own applause,
Of all the Muses that I recollect,
    Whate'er may be her follies or her flaws,
In some things, mine's beyond all contradiction
The most sincere that ever dealt in fiction.
    Byron, Don Juan, Canto XVI, 2