Interactive Modes

These modes are very powerful construction aids. They take full advantage of the mouse. A single click-drag-release sequence constructs several elements. Moreover, the definitions of the new elements are adapted according to the mouse position. For instance, in the "Add a line" mode, the line, together with its start and endpoints, are generated. The definition of the start and endpoint depends on the mouse position when the mouse button is pressed resp. released. So, when the mouse pointer is over an intersection of two lines, then the new point will be defined as the point of intersection of these lines.

If elements are already present in the construction, they will not be added a second time. This refers not only to the definition of the elements, but it is guaranteed by Cinderella's automatic theorem checking facilities.

Interactive modes can easily be recognized in the tool bar since their icon contains a little mouse pointer symbol.

Add a Point

A single new point will be added using this mode usually. The mode is designed to be multi-purpose and easy to use. Pressing the left button of the mouse generates a new point. While you drag the mouse, with the left button still pressed, you can change the position and definition of the newly added point. When you finally release the mouse button the point is fixed. Its definition depends on the position where the mouse was released. Sometimes it can be more convenient to use the more powerful interactive modes ("add a line", "add a parallel", "add a circle", etc.), which generate points together with other geometric elements.

As already mentioned, constructing a point can be described as a three step procedure:

• Pressing the left mouse button generates the point.
• Dragging the mouse moves the new point close to the mouse position. You will recognize that the point snaps to already existing elements. Whenever this happens the definition of the point is adapted to the current situation and the defining elements are highlighted. In particular, the point snaps to intersections of already existing elements, and it snaps to already existing points.
• Releasing the mouse button
  fixes the definition of the point. If at this moment the mouse is ...
  • ... over no element at all, then a free point is added with no additional constraints. In the "move mode" this point can be freely moved.
  • ... over an already existing line, then the new point will always be incident to the line. In the "move mode" this point can only slide along the line.
  • ... over an already existing circle, then a point that is always incident to this circle is added. In the "move mode" this point can only slide along the circle.
  • ... over the intersection of two elements (line, circle, or conic) then the intersection of these elements is added. Also such a point will appear slightly darker than the free points. It is no longer freely movable in the "move mode".
  • ... over an already existing point, then no element will be added.

  The elements that lead to the definition will always be highlighted.

The figures below show the three main situations: a "free point," a "point on a line," and an "intersection point." Notice that while dragging the mouse the coordinates that correspond to the point's position are displayed.

Free point:	Point on a line:	Intersection point:

Clearly a point can also be added with a simple mouse click at the position where the point should be placed. The above mentioned three step procedure collapses then to a single mouse click. However, it may be a matter of taste, it is often more convenient to take the full advantage of the three steps mentioned above:
Press the mouse button down somewhere, drag the point to the right position and release the mouse.

Synopsis

Create a new point with a press-drag-release sequence. The definition is automatically adapted.

Caution

There are two situations where it is not appropriate to add a point with this mode.

• If you want to add an intersection of two lines that cross outside the visible area of the view, then you should use the meet mode.
• If you want to add a point at a position where more than two elements cross, then you should either
  • perturb the situation slightly by using the move mode, if possible,
  • zoom in to get a higher resolution
  • or for line-line intersections use the meet mode to define exactly what intersection is meant.

See also

• Move an element
• Add a line
• Add a circle
• Meet

Add a Line

This mode allows a line through two points to be added. The mode is powerful enough to generate the line together with the two points with just one mouse action. When the mouse is pressed the first point is added. Dragging the mouse generates the second point and the line. When the mouse is released the position of the second point is frozen, and the construction is finished. The logic behind this mode is very similar to the logic behind other interactive modes ("Add a parallel", "Add a perpendicular", "add a circle", etc.).

Constructing a line in this mode is a three step procedure:

• Pressing the left mouse button generates the first point. The definition of this point depends on the position of the mouse at the moment when the button is pressed:
  • If the mouse pointer is over an already existing point, then this point is taken.
  • If the mouse pointer is over the intersection of two elements (line, circle, or conic), then this intersection is automatically constructed and taken as the first point.
  • If the mouse pointer is over just one element (line or circle), then a point is constructed that is constrained to this element. This point is taken as the first point.
  • Otherwise a free point is added.
• Dragging the mouse generates the line and the second point. The definition of the second point is chosen depending on the mouse position. As in the "Add a point" mode the second point snaps to already existing elements. The choice of definitions is completely analogous to the choice for the first point. The elements that define the second point are always highlighted.
• Releasing the mouse button freezes the definition of the second point. The construction is then finished.

The figures below show the three stages during the construction of a line. Here the first point is a free point and the second point is the intersection of the two already existing lines.

Press the mouse ...	... drag it ...	... and release

Synopsis

Create a line connecting two points by a press-drag-release sequence.

See also

• Add a point
• Line through point
• Add a parallel
• Add a perpendicular
• Add a circle
• Join

Line Through Point

This mode creates a line through a point with a certain slope. When the point is moved, then the slope of the line stays constant. However, in the move mode it is also possible to pick the line and change its slope. So this mode could also be called "Line by slope." The mode generates the line together with the point through which it passes by a single press-drag-release sequence. When the mouse is pressed the point is added. Dragging the mouse generates the line. It is always attached to the mouse pointer. When the mouse is released the position of the line is frozen, and the construction is finished. More exactly:

• Pressing the left mouse button generates the point. The definition of this point depends on the position of the mouse at the moment when the button is pressed:
  • If the mouse pointer is over an already existing point, then this point is taken.
  • If the mouse pointer is over the intersection of two elements (line, circle, or conic), then this intersection is automatically constructed and taken as the point.
  • If the mouse pointer is over just one element (line or circle), then a point is constructed that is constrained to this element. This point is taken as the point.
  • Otherwise a free point is added.
• Dragging the mouse generates the line. The slope can be adjusted freely. The line also snaps to already existing points, if they are selected by the current mouse position.
• Releasing the mouse button freezes the definition of the line. The construction is then finished. Depending on the final position of the mouse pointer two things can happen.
  • If the mouse pointer is over an already existing point, then this point is used as a second point on the line. The line is then the join (connecting line) of the first and the second point.
  • Otherwise a "Line with slope" is added.

Synopsis

Create a line through a point by a press-drag-release sequence.

See also

• Add a line
• Circle by radius

Add a Parallel

With this mode you can construct a line through a point which is parallel to another line with a press-drag-release sequence. The point through which the parallel should pass can also be generated in this mode. Constructing the parallel is a three step procedure.

• Move the mouse over the line for which you want a parallel. Press the left mouse button. This creates the parallel and the point through which it should pass.
• Hold the left button and drag the mouse. This moves the parallel and the new point to the desired position.
• Release the mouse. Now the construction is frozen. Depending on the position where you release the mouse the definition of the new point is adapted.
  • If the mouse pointer is over an already existing point, then this point is taken.
  • If the mouse pointer is over the intersection of two elements (line, circle, or conic), then this intersection is automatically constructed and taken as the new point.
  • If the mouse pointer is over just one element (line or circle), then a point is constructed that is constrained to this element. This point is taken as the new point.
  • Otherwise a free point is added.

The figures below show the three stages during the construction of a parallel. Here the new point will be bound to the already existing point P.

Press the mouse ...	... drag it ...	... and release

Synopsis

Create a parallel with a press-drag-release sequence.

Caution

The behavior of this mode is dependent on the geometry that is chosen. While in Euclidean Geometry there is always exactly one parallel, in non-euclidean geometries it is subject to the definition of parallelity. Depending on the underlying "philosophy", in hyperbolic geometry there can be infinitely many parallels, from an incidence geometric viewpoint, or there can be exactly two parallels, from an algebraic or measurement based point of view. Cinderella takes the algebraic point of view: A parallel to a line L is a line that has a zero angle with L. So in hyperbolic geometry the mode produces exactly two parallels.

Hyperbolic Parallels

In elliptic geometry the usual viewpoint is that there are no parallels at all. However, from an algebraic standpoint there exist such parallels, they just have complex coordinates, in other words, they will never be visible. Cinderella constructs these parallels, and they are invisible but are nevertheless present in the "Construction Text" view. From there they can be accessed for further constructions.

See also

• Add a line
• Add a perpendicular
• Define Parallel

Add a Perpendicular

This mode allows a perpendicular to a line and through a point to be constructed with a press-drag-release sequence. The usage of the mode is exactly analogous to the "Add a parallel" mode. The point through which the perpendicular should pass can also be generated in this mode. Constructing the perpendicular is a three step procedure.

• Move the mouse over the line for which you want a perpendicular. Press the left mouse button. The perpendicular and the point through which it should pass are created.
• Hold the left button and drag the mouse. You can move the perpendicular and the new point to the desired position.
• Release the mouse. The construction is frozen. Depending on the position at which you release the mouse the definition of the new point is adapted.
  • If the mouse pointer is over an already existing point, then this point is taken.
  • If the mouse pointer is over the intersection of two elements (line, circle, or conic), then this intersection is automatically constructed and taken as the new point.
  • If the mouse pointer is over just one element (line or circle), then a point is constructed that is constrained to this element. This point is taken as the new point.
  • Otherwise a free point is added.

Synopsis

Create a perpendicular with a press-drag-release sequence.

Caution

The definition of perpendicularity depends on the chosen geometry.

See also

• Add a line
• Add a parallel
• Define Perpendicular

Add a Line With Fixed Angle

This mode allows a line that has a fixed, numerically given, angle to another line to be constructed. The new line and a point through which it should pass can be added with a press-drag-release sequence. The point through which the new line should pass can also be generated with this mode.

When you select this mode, a little window pops up asking you to specify the desired angle.

Input Window:

After that the usage of this mode is analogous to the "Add a parallel" mode.

• Move the mouse over the line to which you want the new line. Press the left mouse button. The new line and the point through which it should pass are created.
• Hold the left button and drag the mouse. You can move the new line and the new point to the desired position.
• Release the mouse. Now the construction is frozen. Depending on the position where you release the mouse the definition of the new point is adapted.

Synopsis

Create a line with fixed angle with respect to another line with a press-drag-release sequence.

Caution:

The definition of angle depends on the chosen geometry.

See also:

• Add a line
• Add a parallel

Add a Circle

This mode allows a circle, given by its center and a point on its perimeter, to be created. The mode generates the circle together with the two points with just one mouse click. When the mouse is pressed the center is added. Dragging the mouse generates the perimeter point and the circle. When the mouse is released the position of the perimeter point is frozen, and the construction is finished. The logic behind this mode is absolutely analogous to the "Add a line" modus.

Constructing a circle in this mode can be described as a three step procedure:

• Pressing the left mouse button generates the center point. The definition of this point depends on the position of the mouse at the moment when the button is pressed, like in the "Add a line" mode.
• Dragging the mouse generates the circle and the perimeter point. Depending on the mouse position the definition of the perimeter point is chosen. As in the "Add point" mode the second point snaps to already existing elements. The choice of definitions is completely analogous to the choice for the first point. The elements that define the perimeter point are highlighted.
• Releasing the mouse button freezes the definition of the perimeter point. The construction is then finished.

Synopsis

Create a circle with a press-drag-release sequence.

Caution

The "shape" of a circle depends on the choice of the geometry.

See also

• Add a line
• Circle by radius
• Circle by fixed radius
• Circle by three points
• Compass

Circle by Radius

This mode allows a circle, given by its center and its radius, to be added. When the center of such a circle is moved, then the radius remains constant. However, in the move mode it is also possible to pick the boundary of the circle itself and change the radius.

The mode generates the circle together with its center with just one press-drag-release sequence. When the mouse is pressed the center is added. Dragging the mouse generates the circle. Its boundary is always attached to the mouse pointer. When the mouse is released the position of the circle is frozen, and the construction is finished. The logic behind this mode is analogous to the "Line through point" modus.

Constructing a circle in this mode can be described as a three step procedure:

• Pressing the left mouse button generates the center. The definition of this point depends on the position of the mouse at the moment when the button is pressed, like in the "Add a line" mode.
• Dragging the mouse generates the circle. The radius can be adjusted freely. The circle also snaps to already existing points, if they are selected at the current mouse position.
• Releasing the mouse button freezes the definition of the circle. The construction is then finished. Depending on the final position of the mouse pointer two things can happen.
  • Either a "Circle by radius" is added,
  • or, if the mouse pointer is over an already existing point, then this point is used as a perimeter point for the circle. The circle is bound to that point.

Synopsis

Create a circle with free radius with a press-drag-release sequence.

Caution

The actual "shape" of a circle depends on the choice of the geometry.

See also

• Add a circle
• Circle by fixed radius
• Line through point

Circle by Fixed Radius

This mode allows a circle that has a fixed, numerically given, radius to be created. The new circle and its center are added with a single press-drag-release sequence.

When you select this mode, a little window pops up asking you to specify the desired radius, similar to the "Line with fixed angle" mode. After that the usage of this mode is analogous to the "Add a point" mode.

Synopsis

Create a circle with numerically given and fixed radius with a press-drag-release sequence.

Caution

The "shape" of a circle depends on the choice of the geometry.

See also

• Add a circle
• Circle by radius
• Add a line with fixed angle

Midpoint

In this mode the midpoint of two points can be constructed with a press-drag-release sequence. Analogous to the "Add a line" mode also the two points can be added.

• Pressing the left mouse button
  ... generates the first point. As in the other interactive modes the definition of this point depends on the position of the mouse at the moment when the button is pressed.
• Dragging the mouse
  ... generates the second point together with the midpoint. The second point also snaps to already existing points or intersections, if they are selected at the current mouse position.
• Releasing the mouse button
  ... freezes the actual definition. The construction is then finished.

Synopsis

Create two points and their midpoint using a press-drag-release sequence.

Caution

This innocent looking mode becomes interesting in non-euclidean geometries. While in Euclidean Geometry there is always a unique, finite, midpoint, in hyperbolic or elliptic geometry there are always two. Both points will be generated in this mode. When working in the Poincaré disc model of hyperbolic geometry, the second midpoint of two points in the disc always lies outside the disc. The presence of this point does not matter a lot, but there are circumstances in which one should be aware of this fact.

See also

• Add a line