Geometria 3.0

La création d'une variable résulte d'une mesure ou d'un calcul. Une variable nouvellement créée a un nom et une définition. Par exemple, la variable maDistance dans l'exemple ci-dessous est définie comme étant la distance entre les points P et W de la figure MonCube :

maDistance = |PW| : MonCube

Dans l'exemple ci-dessous, la variable monAire est le résultat d'un calcul sur des variables monVolume et maDistance préalablement définies :

monAire = 2 * monVolume / maDistance

Les variables nouvellement créées sont automatiquement enregistrées, ainsi que leur définition, sur le calepin.

Les noms des variables peuvent seulement contenir les caractères alphanumériques. Les caractères spéciaux ou accentués ne sont pas autorisés. Un nom de variable ne peut pas contenir d'espace non plus.

Les noms de variables sont sensibles aux majuscules et minuscules. Ainsi, monAire et MonAire feront référence à deux variables distinctes.

PI est un nom réservé à la constante correspondante et ne peut pas être utilisé en tant que nom pour de nouvelles variables. Cependant, il peut être utilisé pour des calculs.

Chaque variable a une valeur numérique. Cependant, dans la plupart des cas, cette valeur reste transparente pour l'utilisateur. Ainsi, l'utilisateur peut créer ou résoudre des problèmes dans Geometria sans jamais connaître la valeur sous-jacente des variables qu'il ou elle a définies. Pour afficher la valeur numérique d'une variable, saisissez le nom de la variable dans la calculatrice et cliquez sur le bouton Evaluer. Dû à des erreurs d'arrondi, les valeurs numériques peut varier légèrement au cours du processus de création ou de résolution d'un problème.

La valeur d'une variable peut changer à la suite de certaines actions sur les figures. Si, dans l'exemple ci-dessus, MonCube subit un agrandissement de coefficient 3, la valeur de maDistance sera implicitement multipliée par 3. Les valeurs de toutes les variables qui dépendent de maDistance, tout comme monAire, seront automatiquement mises à jour également.

Cependant, la valeur numérique de la réponse ne sera pas affectée par des actions sur les figures. Si, dans l'exemple ci-dessus, le rédacteur du problème désigne maDistance comme la réponse au problème avant agrandissement de cette figure, la valeur de la réponse restera inchangée, même si maDistance contient une nouvelle valeur pour les calculs ultérieurs. Cela ne s'applique pas à tous les types de réponse.

Par conséquent, le rédacteur d'un problème devrait envisager de (re)saisir sa réponse juste avant de soumettre ce problème à des résolveurs.

Apres certaines actions, la définition d'une variable peut devenir obsolète. Par exemple, si le point P ou le point W sont effacés, la définition de la variable maDistance ci-dessus devient alors obsolète. Même chose si la figure MonCube est supprimée. Une telle variable est alors dite figée.

Une variable figée n'est plus dépendante de sa définition originale et sa valeur n'est plus mise à jour. Par exemple, effacer le point P dans la figure MonCube va figer la variable maDistance. Si MonCube est ensuite agrandi comme il est décrit ci-dessus, la valeur de maDistance restera néanmoins la même.

Si l'action qui a abouti à figer une variable est annulée, la définition de cette variable est restaurée et la variable est de nouveau mise à jour automatiquement.

Une fois qu'une variable est définie, elle peut être utilisée pour des calculs ou comme argument d'actions comme des mesures, des tracés et des transformations de figures. Une variable peut également être utilisée en tant que réponse d'un problème. A noter qu'une variable figée peut encore être utilisée pour des calculs ou en tant qu'argument.

Une variable existe aussi longtemps qu'elle est enregistrée sur le calepin. Effacer le calepin est la seule façon de supprimer une variable. Soyez prudent car cette action ne peut pas être annulée.

Lorsqu'un document, que ce soit un problème ou une solution, est enregistré, toutes les variables répertoriées sur le calepin sont enregistrées également.

Les calculs sont exécutés à partir d'expressions valides. Les expressions peuvent contenir des nombres, des variables, la constante PI, des opérateurs, des fonctions et des parenthèses.

La calculatrice fonctionne dans le contexte du document ouvert à ce moment-là.

Pour calculer une expression monExpression et assigner le résultat à une nouvelle variable maVariable, saisir :

maVariable = monExpression

et cliquer Evaluer. Le nom de la variable doit être unique dans le contexte du document ouvert. Un nouvel enregistrement va alors être ajouté au calepin. La variable nouvellement créée peut être utilisée dans des calculs ultérieurs et en tant qu'argument.

Si aucun document n'est ouvert, le résultat numérique du calcul sera affiché. Aucune nouvelle variable ne sera créée.

Les opérateurs suivants sont pris en charge :

Les fonctions suivantes sont prises en charge :

Les arguments des fonctions sont indiqués entre parenthèses.

Calculer le côté d'un cube dont le volume est monVolume :

monCôté = monVolume ^ (1 / 3)

Calculer le côté adjacent à l'angle monAngle dans un triangle rectangle dont l'hypoténuse est monHypoténuse :

monCôté = monHypoténuse * cos(monAngle)

The application may behave erratically if the figures contain excessively sized or oddly shaped elements. Here are some guidelines for keeping the figures safe:

Problems, solutions and figures created with Geometria versions prior to 3.0 are no longer supported. However, we plan to have all future versions of Geometria backward compatible with 3.0.

To define the answer to the currently open problem, select Answer from the Edit menu. Select the answer type:

Open the Value pane.

The solver concludes the solution by providing a correct answer. Note, again, that the solver's answer is considered correct if it matches the creator's.

Have the solution opened. Select Answer from the Edit menu. Enter a valid expression or comma-separated labels, as appropriate. To minimize the input, have a notepad record and/or some elements, such as points, segments or a face, selected ahead of time in the selected figure. Click OK.

An evaluation of the answer (corect or incorrect) will pop up.

If the answer is correct, it will conclude the solution log. No more actions will be possible unless the answer is undone. You may want to save the solution to a file. A saved solution can later be played back.

See also:

Geometria employs a fixed right-hand reference system. The Z axis of the reference system is parallel to the vertical edge of the screen. No knowledge of the reference system is necessary to manipulate figures, create and solve problems.

At any given time, the figure has a certain attitude. The attitude can be described in terms of angles the figure forms with the reference system. When a figure is created, it has the default attitude. The figure's attitude changes as the figure is rotated.

To revert a figure to the initial attitude, select Initial Attitude from the View menu or right click and select Initial Attitude from the popup menu.

When a figure is saved, its current attitude becomes its initial attitude. To revert to the attitude the figure had when it was created, before any subsequent rotations, select Default Attitude from the View menu.

See also:

Toggle the selector off and stroke a figure with the mouse to rotate the figure. The rotation speed and angle will depend on the stroke.

To pin a rotating figure, click on it.

See also:

To zoom in on a figure, select Zoom In from the View menu or click the button on the tool bar.

To zoom out, select Zoom Out from the View menu or click the button on the tool bar.

To resize a figure so it fits into the window, select Fit To View from the View menu or click the button on the tool bar.

See also:

To measure the distance between two points, proceed, alternatively, as follows:

The two points must belong to the same face.

A dialog will pop up. Fill in, if necessary, the points, assign a unique variable and click OK. A record will appear on the notepad, as below:

myDistance = |AB| : MyCube

where myDistance is the variable assigned to the distance between points A and B in figure MyCube. If MyCube is currently the only figure, the record will read:

myDistance = |AB|

This action can be undone.

See also:

To measure an angle, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog will pop up. Fill in, if necessary, the points, assign a unique variable and click OK. A record will appear on the notepad, as below:

myAngle = <ABC : MyPyramid

where myAngle is the variable assigned to the angle between AB and AC in figure MyPyramid. If MyPyramid is currently the only figure, the record will read:

myAngle = <ABC

All angles are expressed in radians.

This action can be undone.

See also:

To measure the area of a face proceed, alternatively, as follows:

A dialog will pop up. Select, if necessary, the face from the list, assign a unique variable and click OK. A record will appear on the notepad, as below:

myArea = area[ABC] : MyCube

where myArea is the variable assigned to the area of the face that contains vertices A, B, C in figure MyCube. If MyCube is currently the only figure, the record will read:

myArea = area[ABC]

This action can be undone.

See also:

To measure the volume of a figure proceed, alternatively, as follows:

A dialog will pop up. Assign a unique variable and click OK. A record will appear on the notepad, as below:

myVolume = volume : MyCube

where myVolume is the variable assigned to the volume of figure MyCube. If MyCube is currently the only figure, the record will read:

myVolume = volume

This action can be undone.

See also:

To join two points with a segment, proceed, alternatively, as follows:

The two points must belong to the same face.

If a dialog pops up, fill in the points and click OK.

This action can be undone.

See also:

To draw a perpendicular through a point to a segment, proceed, alternatively, as follows:

The point and the segment must belong to the same face. The point may or may not belong to the segment.

If a dialog pops up, fill in the points and click OK.

If the point and the segment belong to two different faces at the same time, a second dialog may pop up. Select a face from the list and click OK.

This action can be undone.

See also:

To divide a segment in a given ratio, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as AX : XB, where X is the division point, AB the segment. For example, if the ratio is 1 : 2, X will be twice closer to A than to B. If the ratio is 3 : 1, X will be three times closer to B than to A.

The midpoint divides the segment in ratio 1 : 1 and therefore can be drawn as shown above. However, a convenience Draw Midpoint action offers a shortcut.

This action can be undone.

See also:

To draw the midpoint of a segment, proceed, alternatively, as follows:

A dialog may pop up. Fill in, if necessary, the points. Click OK.

Note that drawing the midpoint is equivalent to dividing the segment in ratio 1 : 1.

This action can be undone.

See also:

To divide an angle in a given ratio, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog will pop up. Fill in, if necessary, the points and valid expressions for the numerator and denominator of the ratio. Click OK.

The ratio is defined as <AOX : <XOB, where OX is the division segment, <AOB the angle. For example, if the ratio is 1 : 2, OX will make with OB a twice greater angle than with OA. If the ratio is 3 : 1, OX will make with OA three times greater an angle than with OB.

The bisector divides the angle in ratio 1 : 1, therefore can be drawn as shown above. However, the convenience Draw Bisector action offers a shortcut.

This action can be undone.

See also:

To draw the bisector of an angle, proceed, alternatively, as follows:

The segments that make the angle must belong to the same face and not be collinear.

A dialog may pop up. Fill in, if necessary, the points. Click OK.

Note that drawing the bisector is equivalent to dividing the angle in ratio 1 : 1.

This action can be undone.

See also:

To draw the intersection point of the lines containing two given segments, proceed, alternatively, as follows:

Both segments, as well as the intersection point of their lines, must belong to the same face.

A dialog may pop up. Fill in, if necessary, the points. Click OK.

This action can be undone.

See also:

To draw a point on a given segment at a given distance from a given point, proceed, alternatively, as follows:

The point and the segment must belong to the same face. The point may or may not belong to the segment.

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the distance. Click OK. Select a notepad record ahead of time to have it copied automatically to the distance input box.

This action may result in the construction of one or two distinct points.

This action can be undone.

See also:

To draw a segment from the end point of another segment so that the two segments make a given angle, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the angle. Click OK. Select a notepad record ahead of time to have it copied automatically to the angle input box.

The angle must be greater than 0 and less than 180 degrees and be entered in radians, as opposed to degrees. For example, to lay an angle of 60 degrees, enter PI/3.

If the segment belongs to two different faces at the same time, a second dialog may pop up. Select a face from the list and click OK.

This action may result in the construction of one or two distinct segments.

This action can be undone.

See also:

The problem creator can rename a point in a figure by selecting Rename Point from the Draw menu.

A dialog will pop up. Enter the old and the new labels for the point in question and click OK. A valid label consists of a capital letter, which may be followed by one or more digits.

Select a point ahead of time to have it automatically copied into the input box.

Renaming a point may result in some of the notepad records being updated accordingly.

This action can be undone.

See also:

To scale a figure along a given segment by a given positive factor, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the points and a valid expression for the factor. Click OK. Select a notepad record ahead of time to have it automatically copied to the factor input box.

As a result of scaling, the previously measured distances, angles, areas and volumes will be recalculated and the assigned variables updated. See Variables for more details.

This action can be undone.

The solver cannot scale a figure that is part of the problem. However, the figures that have been created in the process of solving the problem may be scaled.

See also:

Shearing refers to an affine transformation where planes parallel to a fixed reference plane slide along a fixed line contained in the reference plane. The term 'affine' implies that the transformation preserves the collinearity of points.

If XY is the fixed line in the reference plane, the shearing transformation is uniquely identified by points X, Y and the image of any point Z such that XZ is perpendicular to the reference plane.

To shear a figure, proceed, alternatively, as follows:

A dialog will pop up. Fill in, if necessary, the reference points (X, Y at the bottom and the image of point Z at the top). Click OK.

As a result of shearing, the previously measured distances and angles will be recalculated and the assigned variables updated. See Variables for more details.

This action can be undone.

The solver cannot shear a figure that is part of the problem. However, the figures that have been created in the process of solving the problem may be sheared.

See also:

To cut a figure with a plane, proceed, alternatively, as follows:

The selected elements must uniquely identify a plane. The plane must pass through the figure's interior.

If a dialog pops up, fill in the points and click OK.

The two resulting halves will be displayed in separate views. The original figure will be kept intact in its own view.

This action can be undone.

See also:

Two figures can be joined by either selecting Join Figures from the Transform menu or by clicking the button on the tool bar.

A dialog will pop up. Select the figures to join and click OK. A figure can be joined with a copy of itself. Therefore the same figure can be selected in both lists.

If the figures can be joined in one way only, a new view will appear with the combined figure in it. The two original figures will be kept intact in their own views.

If, however, the figures can be joined in more than one way, another dialog will be displayed asking if the figures should be joined in any of the possible ways. Click No to proceed with a more accurate definition of the joining elements.

Click Yes to have the figures joined in one of the possible ways. A preview will be displayed along with another dialog asking if the result should be kept. Click Yes to keep the result and conclude the action. Click No to get a preview of the next possible way of joining the figures.

It is recommended that, prior to joining the figures, the corresponding points and/or faces be selected. That will reduce to a minimum the subsequent dialog input. Note that faces can only be selected in the opaque view mode.

The joining faces must have the same shape, that is have equal angles and proportional sides at corresponding vertices. The joining faces do not have to have the same size. If need be, one of the figures will be automatically scaled to fit the other.

The action will be cancelled if the joint figure is not convex.

This action can be undone.

See also: