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# «Geometry Expressions Manual TM Saltire Software PO Box 230755 Tigard, OR 97281 0755 © 2013 Saltire Software. All rights reserved. Information in ...»

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3. If you choose "Relax other constraints so the coordinate is independent", this will keep the new constraint you just entered and allow you to eliminate one of the red highlighted constraints (figure 1).

When you select one of these constraints (in figure 2 below we clicked on q), the highlight changes to gray. After you click Ok, the selected constraint, the angle q in this case, is calculated and displayed (figure 3).

–  –  –

You can click the constraint and drag it to adjust its placement on the drawing.

To specify the radius of a circle:

–  –  –

well as the icon.

2. Click the Radius icon, enter the constraint value, either real or symbolic, and press enter. You can press enter without typing a value to let the system insert a variable name.

You can click the constraint and drag it to adjust its placement on the drawing.

Perpendicular Constraint Any two of lines, segments, vectors or polygon sides can be constrained to

be perpendicular with these steps:

1. Select two from the line types listed above.

–  –  –

Angle Constraint Any two of lines, segments, vectors or polygon sides can be constrained

with an angle value or variable name with these steps:

–  –  –

Which Side to Constrain?

Sometimes when identifying angles, the constraint falls on the wrong one.

In the example below, we wanted BDC, not BDA. Just click the cursor over the constraint arrow and drag it to the other side, then release the mouse button - done!

Note: The angular units are displayed in the lower right of the screen.

Change the default Angle Mode in the Edit / Preferences / Math Properties menu, Math settings group.

© 2013 Saltire Software 86 Geometry Expressions Manual Direction Constraint Constrain any of the line types; line, line segment, vector, or polygon side, to a direction measured from the horizontal.

1. Select one of the line types listed above.

2. Click the Direction icon.

3. Enter the constraint, real or symbolic. If you enter a real value, the line will be adjusted to reflect the constraint.

Note: The angular units are displayed in the lower right of the screen.

Change the default (Degrees or Radians) in the Edit / Preferences menu.

Slope Constraint Specify a slope for any of the line types; line, line segment, vector, or polygon side.

1. Select one of the line types listed above.

2. Click the Slope icon.

3. Enter the constraint, real or symbolic. If you enter a real value, the line will be adjusted to reflect the constraint.

–  –  –

2. Click the Coordinate icon.

3. Enter the constraint, real or symbolic. If you enter a real value, the line will be adjusted to reflect the constraint, even if the coordinate axes are not displayed.

To change the coordinates shown, double click and type over the highlighted value in the data entry box.

Constraining Vector Coeffecients

You can specify coefficients for a vector with the following steps:

–  –  –

2. Click the coefficients icon.

3. Enter the coefficients separated by a comma.

Note: Don't forget the parentheses or an error message appears.

© 2013 Saltire Software 88 Geometry Expressions Manual Tangent Constraint Any of the line types; line, line segment, vector, or polygon side can be

made tangent to a circle or locus with these steps:

1. Select a line of the types listed above and the circle or locus.

2. Click the Tangent icon from the Constrain tool box or select Tangent from the Constrain menu.

The line and curve immediately become tangent.

Incident Constraint Constrain a point to be incident to any other geometry; line, segment,

vector, polygon side, circle or locus with these steps:

1. Select the point and the other geometry listed above.

2. Click the Incident icon from the Constrain toolbox, or select Incident from the Constrain menu.

The point is moved to meet the line or curve, or the extension of the line.

Below is an example of the latter, point D is moved to lie on the extension of line segment AB.

–  –  –

If you select the point or the line, incidence is indicated by a bowtie

around the point:

Congruent Constraint Constrain two or three of any of these geometry types: line segments,

vectors, or polygon sides, to be congruent with these steps:

–  –  –

2. Click the Congruent icon from the Constrain toolbox, or select Congruent from the Constrain menu.

© 2013 Saltire Software 90 Geometry Expressions Manual You will see matching congruency lines on the selected segments and a length will be adjusted.

Parallel Constraint Any two or three of the linear geometry types can be made parallel: line, segment, vector, or polygon side.

1. Select two or three from the types listed above.

2. Click the Parallel icon from the Constrain toolbox, or select Parallel from the Constrain menu.

The geometry will be adjusted and matching symbols

–  –  –

Implicit Equation Constraint You can use symbolic variables to constrain geometry with an implicit equation. Lines, line segments, polygon sides, vectors and circles and conics can all be constrained with implicit equations.

–  –  –

2. Click the Implicit Equation icon from the Constrain toolbox, or select Implicit Equation from the Constrain menu.

An input window will open next to the geometry you selected. Highlighted in the window is a generic equation for the selected object; for a line, an equation like - XA1+YB1+C1 = 0 might appear. You can edit the equation with different variable names or coefficients as you like. You will find these variables added to the variable list in the Variables toolbox.

Point Proportional Along a Curve Constraint

A point proportion t along a curve is defined variously for different types of

curves as follows:

For a Line segment AB, it defines the point (1-t)•A + t•B · For a Circle it defines the point on the circle which subtends angle t at · the center.

For a Locus or envelope, it defines the point at parameter value t.

· For general Cartesian functions, it defines the x value of the point on · the function.

© 2013 Saltire Software 92 Geometry Expressions Manual

For Polar functions, it defines the point on the function which · subtends angle t.

For general Parametric functions, it defines the point at parameter · value t.

For an Ellipse of the form X2/a2 + Y2/b2 =1 it defines the point (a cos( · t), b sin(t)).

For a Parabola of the form Y=X2/4a it defines the point (2at, at2) · For a Hyperbola of the form X2/a2 - Y2/b2 =1 it defines the point (a/ · cos(t), (b sin(t))/cos(t)).

–  –  –

2. Click the Point Proportional icon from the Constrain toolbox, or select Point Proportional from the Constrain menu.

3. Enter the parameter or quantity (symbolic or real) in the data entry box.

For example, in the following diagram, D is defined proportion t along AB, and E is defined proportion t along BC. The curve is the locus of F as t varies between 0 and 1.

–  –  –

In the following example, the curve is the locus of the point (x,x2).

Tangents are created at points with parameter values x0 and x1 on this curve.

© 2013 Saltire Software 94 Geometry Expressions Manual Where is Point proportional along curve for conics?

The best way to understand the location of Point proportional along curve command for conics is to see how we construct it geometrically for

each conic:

Ellipse The ellipse with foci A and B is inscribed in circle, center M. Draw the radius MN at angle t to the major axis and drop the segment NO perpendicular to the major axis of the ellipse. When the intersection of NO with the ellipse (point C) is constrained to be t proportional along the ellipse, it's coordinates will be (a cos(t), b sin(t)).

© 2013 Saltire Software Tools 95 Parabola C lies on the parabola and BC is perpendicular to the axis AB of the parabola. Point D is located proportion t along the segment. Point F is the intersection of the perpendicular to BC through D with the parabola. It has the coordinates (2at, at2) when it is constrained to parametric location t on this parabola.

Hyperbola CD is the perpendicular projection of C onto the axis of the hyperbola, GF is the circle centered at the center of the hyperbola which goes through the intersections of the hyperbola with its axis. H is the point of contact of this circle with the tangent from D. We can see that the angle DGH is the same as the parameter value. When point C is constrained to be at parametric location t along the curve, its coordinates are (a/cos(t), b sin(t )/cos(t)) on this hyperbola.

© 2013 Saltire Software 96 Geometry Expressions Manual Constructions Creating Constructions After sketching and constraining your drawing there are a whole set of constructions that can be applied to the geometry. First you must select the geometry elements which pertain to the construction. When you select the geometry the appropriate constructions will be highlighted.

The following table lists the Constructions, their icons, and which elements must be preselected to activate the constructions. Be careful when selecting geometry objects, if extra things are selected that are not related to the construction, the construction icons will remain inactive.

This can happen by mistake, especially when using the selection box tool.

–  –  –

Midpoints of Line Segments You can construct a midpoint on any line segment, vector, polygon side, or

between two points by:

1. Select two from the geometry types listed above.

–  –  –

A point will appear in the middle of the selected line.

Intersections You can construct a point of intersection between any of the line types in your geometry; line, segment, vector, polygon side or circle. You can also construct intersections of circles. Conics are limited to intersections only with lines, segments or vectors.

1. Select two from the line types listed above.

2. Click the Intersection tool or select Intersection from the Construct menu.

–  –  –

A new point and label will appear at the intersection. If the lines are segments that do not intersect, a point will be created at the extension of the lines as with line segments AB and CD below.

If the geometry will never intersect, the selected objects are moved to form the intersection. In the example below, the infinite line and circle become tangent at the newly created point, H.

Perpendicular Bisector You can construct a perpendicular bisector on any line, segment, vector,

or polygon side with these steps:

–  –  –

An infinite line will appear at right angles to the selected line.

© 2013 Saltire Software 100 Geometry Expressions Manual Angle Bisector You can bisect the angle between any combination of line types; line,

segment, vector, or polygon side with these steps:

1. Select two of any of the line types listed above.

2. Click the Angle Bisector tool or select Angle Bisector from the Construct menu.

An infinite line will appear between the two selected lines. You can use the Calculate / Angle tool to get the value of the bisected angle.

Parallel Constructions You can construct a line, through a point, and parallel to another line,

segment, polygon side or vector with these steps:

1. Select a point and a line of one of the types listed above.

2. Click the Parallel tool or select Parallel from the Construct menu.

A line is constructed which is parallel to the selected line and passes through the selected point.

–  –  –

Perpendicular Constructions You can construct a line, through a point, which is perpendicular to

another line, segment, polygon side or vector with these steps:

–  –  –

A line is constructed which is perpendicular to the selected line and passes through the selected point.

Tangents You can construct a line that is tangent to a circle or curve with these

steps:

1. Select the circle or curve. You can also select a point on the curve so that the tangent goes through the point on the curve.

–  –  –

A line tangent to the selected curve will appear at the point where you selected the circle or curve, or at the selected point.

© 2013 Saltire Software 102 Geometry Expressions Manual Polygon Construction If you created a polygon with the line segment tool, or your polygon was not shaded for some reason, (e.g. the drawing of the sides was interrupted or out of order) you can make joined line segments into a polygon that can be selected with a single click using this construction.

1. Select the line segments that make up the polygon.

2. Click the Polygon tool in the Construct toolbox, or select Polygon from the Construct menu.

The polygon will be filled and you can now select the entire polygon with a single click.

Reflection

You can reflect any subset of your diagram about a line with these steps:

1. Select one or more geometry objects to reflect.

2. Click the Reflection tool in the Construct toolbox, or select Reflection from the Construct menu.

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