Tools for general conics

You access the conics-tools by pressing the button or the drop arrow on its right:

[conics_menu]

To read here you should be familiar with Generalities on tools

[bullet] Conic characterisitics

By conics we mean usually the so called proper conics, which are ellipses, parabolas , hyperbolas . These are curves represented by equations of degree two:
f(x,y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0.
These equations however allow for the representation of some other curves called improper conics:
a) The union of two lines, when the equation can be written as a product (hx + ky + l)(mx + ny + p)=0.
b) A single line, when in the above equation a = b = c = 0 .
c) A double point, when the above equation can be written in the form (x-r)2 + (y-s)2 = 0.
Most of the time we are concerned with proper conics. For some tools however we must consider also the improper ones. The circle is a special conic (an ellipse with coinciding focal points), but in this program it is handled as a distinguished kind (different from an ellipse).

[bullet] More (interpolation) points

Menu-item: Conics General \ More points _
Keyboard shortcut:   None
Description
Operates in one selective stage. Simply click on a proper conic to augment its interpolation points by 60.
Remarks
a) Each time you click on a proper conic the tool adds to it 60 additional interpolation points, making the drawing more precise. Don't exaggerate however, else you receive a penalty on speed.
b) If you hold down the CTRL key, while you click on the proper conic, then 60 points are removed, instead to be added. This operation takes place only if the remaining points to be left are more than 20.

[bullet] Focal points

Menu-item: Conics General \ Focal points _
Keyboard shortcut:   None
Description
Operates in one selective stage. Simply click on a proper conic to define its focal points. The tool creates a group of two points and a dependence relation of type kConicFoci.
Remark
This tool is useful for conics generated as family-members from other conics. In such cases the conic is easily calculated, but its foci are not automatically drawn.

[bullet] Principal axes

Menu-item: Conics General \ Principal axes _
Keyboard shortcut:   None
Description
Operates in one selective stage. Simply click on a proper conic to define its (orthogonal) principal axes. The tool creates a group of two lines and a dependence relation of type kConicAxes.
Remark
In most cases the axes of a proper conic are not automatically drawn. So use this tool if you need them.

[bullet] Directrix

Menu-item: Conics General \ Directrix _
Keyboard shortcut:   None
Description
Actually the tool should be called Directrices, since there are two of them. The tool creates a group of two parallel lines and a dependence relation of type kConicDirectrix.
Remark
The parabola has only one directrix. The program however creates also in this case a group of two identical lines. By this we avoid extra calculations when the parabola is modified somehow and falls into the case of ellipses or hyperbolas. It doesn't harm and the user sees only one line, as it should.

[bullet] Conjugate Diameters

Menu-item: Conics General \ Conjugate Diameters _
Keyboard shortcut:   None
Description
The tool operates in two stages:
stage1(A,B) is selective. It selects at A a proper conic.
stage2(C,D) is constructive. It creates a point X and a group of two segments. This group is a dependent of the conic and the point characterized by the symbolic constant kConicConjDiam. The masters of the group are the conic and the point. The point is used to determine the first of the diameters. The line of this diameter contains the point. The other diameter is the conjugate of the first one.
Remarks
a)   The point, determining the first diameter, can be moved so that the first diameter matches a certain direction. You can make it invisible if its presence is not desirable.
b)   If you want the diameter at a precise point of the conic, set the point on the conic before to use this tool and then click for X next to this point.

[bullet] Dual Conic

Menu-item: Conics General \ Dual conic _
Keyboard shortcut:   None
Description
The tool operates in one selective stage constructing the dual conic of the proper conic selected.
The conic created is a dependent of the selected one. The dependency is characterized by the symbolic constant kConicDual.

[bullet] Normal osculating circle of a Conic at a point

Menu-item: Conics General \ Normal/Osculating at _
Keyboard shortcut:   None
Description
The tool operates in one selective and one constructive stages:
stage(A,B) selects a proper conic at A, say X.
stage(C,D) sets a point at D, say Y, and projects it to the nearest point Z on the conic. Then it constructs the osculating circle W of the conic at Z. The circle W is a dependent of the conic X and the point Y (masters of W). No parameters are needed. The dependency is characterized by the symbolic constant kOsculating.
Remark
The point Y could be considered as the pivot, defining by its projection on the conic, the point at which the normal is considered. If you want the osculating circle at a precise point of the conic, set the point on the conic before to use this tool and then click for Y next to this point.

[bullet] Conic tangent to five lines

Menu-item: Conics General \ Tangent 5 lines conic
Keyboard shortcut:   None
Description
The tool operates in 5 selective stages: stage1(A1,B1) ... stage5(A5,B5). In each stage it selects a side/segment/line s1,...,s5 of a corresponding object o1,...,o5 (can be identical f.e. the 5 sides of one pentagon).
In the last stage the conic Y tangent to the 5 lines is constructed as a dependent object Y of type kConic5Tan.
Masters
The masters are the different among the five o1,...,o5. .
Parameters
The parameters are a sequence of 5 pairs of integers m1,n1,...,m5,n5. The m's show which master on the list of o1,...,o5 and the n's show which side of that master has been selected.

[bullet] Conic passing through five points

Toolbar-button:[conic5]
Menu-item: Conics General \ Conic through 5 points
Keyboard shortcut:   None
Description
The tool operates in 5 constructive stages: stage1(A1,B1) ... stage5(A5,B5), creating 5 points P1,..,P5.
In the last stage the conic Y, passing through the 5 points is constructed as a dependent object Y of type kConic5Pts.
Masters
The masters are the 5 points P1,...,P5 (must be different). There are no parameters

[bullet] Level-Conic passing through a point

Menu-item: Conics General \ Level-Conic _
Keyboard shortcut:   None
Description
The tool operates in one selecting stage1(A1,B1) and one constructive stage2(A2,B2).
In the first stage it selects a conic X at A1 and in the second sets a point Y at B2. Then it creates a conic passing through Y and having the same quadratic equation with that of X, except for its constant term. The conic created is a dependent object having as masters X, and Y and no parameters. Its type is kConicLevel.
Remark
If the conic X is a hyperbola, then the level conics are all hyperbolas with the same asymptotes. The conjugate hyperbola, as well as the two asymptotes are included (as a degenerate conic) in the set of level curves.

[bullet] Matrix of the conic

Menu-item: Conics General \ Its Matrix _
Keyboard shortcut:   None
Description
The tool operates in one selecting stage(A,B) selecting a conic X and constructing a text-box containing 3 matrices.
The first matrix is the symmetric matrix of coeficients of the conic. The second is the matrix, reducing by similarity the first one to a diagonal matrix. The last matrix is this diagonal matrix.
The text-object is a dependent of the conic of type kText.
Parameters
There are two integer and two real (double) parameters. The first is the value of the constant kConicMatrix the second is the (0-based) count of the anchor to which the text-box is attached. The two last parameters determine the position of the top-left corner of the text-box.

[tangential family]

 

You access the conics-families-tools by pressing the drop arrow:

[fconics_menu]

[bullet] Families of conics

Families of conics are defined as lines in the space of all conics and usually have the form
(1-k)f(x,y) + kg(x,y) = 0.
Here f, g are two quadratic equations and each value of k gives a particular conic. Since f, g can represent (double) points, lines, pairs of lines circles and proper conics, interesting combinations result. This is the list of the cases considered, each giving a particular kind of family (M. Berger's Geometry, Springer, paragraphs 16.4 and 16.5 contains the classification of the complex case). Each family containing non-degenerate conics, contains also at least one and at most three degenerate conics.
One could generate most of the families by combining their degenerate members, thus reducing the list below to a smaller one that generates all cases. For the sake of convenience though I left them all alive. In the list below I put in parentheses the symbolic constant characterizing the particular family.
    Conic-family selectors
  1. Point and Point (kPntPnt) --> generates an hyperbolic bundle of circles with these base points (image)
  2. Point and Line1 (kPntLn1) --> as before with Point as one base point and Line1 as radical axis of the bundle
  3. Point and LineD (kPntLnD) --> generates conics with the LineD as directrix and the Point as focus (image)
  4. Point and Line2 (kPntLn2) --> generates mainly hyperbolas with ellipses concentrating around the point (image)
  5. Point and Circle (kPntCir) --> generates an hyperbolic circle-bundle with base-pts at the Point and its inverse to the Circle
  6. Point and Conic (kPntConic) --> generates the same families of conics with case 3 (Point and Line2)
  7. LineD and Circle (kPntLnD) --> like in 2 (Point and LineD) with focus the pole of the line w.r. to the circle
  8. LineD and Conic (kLnConic) --> generates a family containing all sorts with ellipses concentrating around the pole of LineD w.r. to the conic (image)
  9. Circle and Circle (kCirCir) --> generates the circle bundle of the two circles
  10. Circle and Conic (kCirConic) --> generates a family of conics containing all sorts of conics and only one circle
  11. Conic and Conic (kConConic) --> most general case generated by proper conics, including all others as particular cases (image1) , (image2)
  12. Line2 and Circle (kLn2Cir) --> similar to the case Point and Line2 contains a single circle as member
  13. Line2 and Conic (kLn2Conic) --> similar to previous, the Line2 is a singular member of the family generated
  14. Line2 and LineD (kLn2LnD) --> if LineD intersects Line2 at A, B, then all members are tangent at A and B (image)
  15. Line2 and Line2 (kLn2Ln2) --> generates all conics through the four intersection points of the lines (image)
  16. Line1 and Circle (kLn1Cir) --> circle bundle with Line1 as common radical axis
  17. Line1 and Conic (kLn1Conic) --> generates a family of conics of the same kind (f.e. all ellipses) except the Line1 (image)
  18. Line1 and LineD (kLn1LnD) --> generates a family of parabolas tangent to Line1 at its intersection with LineD which is parallel to the parabola's axis (image)
  19. Line2 and Line1 (kLn2Ln1) --> if A, B the intersections of Line2, Line1, all members are hyperbolas through A, B (image)
Line1 means a single line
LineD means a line considered double (its equation to power 2)
Line2 means a set of two different lines (intersecting or parallel)

[bullet] Two conics family

Menu-item: Families of Conics \ 2-Conics Family _ _
Keyboard shortcut:   None
Description
The tool operates in two selecting stages and one constructive. The first two select two conics X, Y and the last sets a point P through which we want the family-member to pass. In the last stage the tool creates a conic passing through P, depending on X, Y and P in a dependence characterized by the symbolic constant kConicFamily.
Parameters
There is one integer identifying the pair-kind of the objects. f.e. 7 stands for a pair of selected objects X, Y, where one of them is a line (considered double) and the other is a proper conic. The integers representing the various cases are the list-counts of the previous list.
Remarks
a) If in some selection stage you select a line, the line is considered automatically double (LineD notation in the list above). If you wish to interpret the line as a single (line1 in the list), hold down the shift key while clicking on it and in addition hold down the shift key when setting a point to define the family member through that point.
b) With this tool you handle all the cases which need the selection of two distinguished objects. The other two tools for families handle the cases where you need three and four objects to select.

[bullet] Conic families generated by two lines and a conic

Menu-item: Families of Conics \ 2 Lines + conic _ _ _
Keyboard shortcut:   None
Description
The tool operates in three selecting stages and one constructive.
The first two select two different lines X, Y.
The next, third selective stage, selects a conic Z (may be proper or not).
The last, fourth stage is constructive and sets a point P through which we want the family-member to pass. In the last stage the tool creates a conic passing through P, depending on X, Y, Z and P in a dependence characterized by the symbolic constant kConicFamily.
Parameters
There is one integer identifying the pair-kind of the objects. The integers representing the various cases are the list-counts of the previous list.

Remarks
a) If in the third selection stage you select a line, the line is considered automatically double (LineD notation in the list above). If you wish to interpret the line as a single (line1 in the list), hold down the shift key while clicking on it and in addition hold down the shift key when setting a point to define the family member through that point. The remark concerns the distinction of the cases 13 and 18 in the above list.
b) Notice that this tool does not define additional families not constructible by the previous tool. It simply gives an other way (using their degenerate members) to create some of these families. The example contains a family generated by two lines a, b (a Line2 in our symbols) and a point B. The same family is generated by the point B and an ellipse-member c.

[bullet] Conic families generated by two (degenerate conics) pairs of lines

Menu-item: Families of Conics \ 2Lines + 2Lines _ _ _ _
Keyboard shortcut:   None
Description
The tool operates in four selecting stages and one constructive.
The first two select two different lines X, Y.
The next select two other different lines U, V.
The last, fifth stage is constructive and sets a point P through which we want the family-member to pass. In the last stage the tool creates a conic passing through P, depending on X, Y, U, V and P in a dependence characterized by the symbolic constant kConicFamily.
Parameters
There is one integer identifying the pair-kind of the objects. The integers representing the various cases are the list-counts of the previous list. For this tool the corresponding value is always 14.
Remarks
a) Except the two defining degenerate conics of the family, there is an additional family member which is also degenerate conic consisting of two lines (Line2 in our notation). The example contains a family generated by two degenerate conics and showing the third degenerate member.
b) Notice that all family members pass through the four intersection points of the two defining pairs of lines.

[bullet] Singular members of the family of conics

Menu-item: Families of Conics \ Singular Members _ _
Keyboard shortcut:   None
Description
The tool operates in two selecting stages.
The first two select two different proper conics X, Y.
Right after the selection of the second conic, the tool searches for the degenerate (singular) conics contained in the family. These are either a pair of lines (intersecting, parallel, double) or some (double) points. The tool creates a group of 6 objects (lines or/and points), depending on the two conics. There are no parameters and the dependence relation is characterized by the symbolic constant kSingularConic.
Remarks
a) In the case of points, they are considered as double and represented by two identical point-objects.
b) No family other than a hyperbolic circle bundle can contain more than one double point.

There's only one slight difference between
    Me and my epic brethern gone before,
And here the advantage is my own, I ween;
    (Not that I have not several merits more,
But this will more peculiarly be seen)
    They so embellish, that 'tis quite a bore
Their labyrinth of fables to thread through,
Whereas this story's actually true.
    Byron, Don Juan, Canto I, 202