Euklides

EUKLIDES

Euklides is an euclidean geometry drawing language. The compiler euklides allows you to typeset geometry figures within a (La)TeX document. The X-Window program xeuklides makes possible to create interactive geometry figures.

This is version 1.3 of Euklides documentation.

Overview of the Euklides system		An overview of the Euklides system
1. Design of the language		The design of the language
2. Handling numbers and number valued functions
3. Handling vectors
4. Point valued functions
5. Line valued functions
6. Segment valued functions
7. Circle valued functions
8. Geometrical transformation functions
9. Defining triangles
10. Defining polygons
11. Intersection points		Determining intersection points
12. Interactive assignements		Defining interactive variables
13. General graphical commands
14. Drawing commands
15. Samples		Some source files of classical figures
Contact		Bug reports and comments
Concept index		Index of concepts
Command index		Index of commands

Overview of the Euklides system

The language Euklides provides simple and powerfull commands for creating elementary euclidean geometry figures. The underlying philosophy is to avoid as much as possible the use of coordinates.
The compiler euklides translates a source file written in this specific language into TeX (pstricks) macrocommands. (The result is sent to standard output.)
The X-Window program xeuklides is a complete system for editing and viewing figures described in this language. With xeuklides, the figures can be interactive, i.e., while viewing, it is possible to modify some numerical values using the keyboard.

There are three situations where Euklides can be usefull:

• You want to create a geometry figure in EPS format. In this case you can run the shell script euk2eps on a Euklides source file.
• You want to insert some geometry figures in a TeX document. In this case you can insert some Euklides commands within your TeX source file. You must enclose them between the following special comments: %--euklides at begin and %--end or %--stop at end. The %--stop ending tag tells euklides to keep variables in memory for further use. With the %--end tag, variables are cleared. If you run euklides with the -f option on the source file, the Euklides commands are replaced with pstricks commands and you obtain a ready-to-tex file. (Don't forget to load the pstricks package.)
• You want to create an interactive geometry figure. In this case you can run xeuklides.

1. Design of the language

An Euklides source file can contain:

• Comments, i.e. any text beginning with `%' and ending at the end of the line.
• Simple assignments, i.e. a variable name followed by `=' and an expression.
• Multiple assignments, i.e. a variable name list followed by a multiple assignment command (triangle assignment, polygon assignment or intersection assignment).
• Graphical commands, i.e. parameter setting commands or drawing commands.

A variable name is a letter, possibly followed by any letter or digit. Warning: Euklides is case sensitive. An expression can have several type of values : number, vector, line, segment, circle. A line of the source file can contain a unique assignement or command. It can also contain several ones, in this case they must be separed by semicolons. (No semicolon at the end of the line.)

We'll use the following notations (possibly followed by a digit) to describe the possible types of the parameters in a command:

• x, y, z: number.
• ang : angle measure.
• A, B, C, D, E, F, G: point.
• u, v: vector.
• l: line.
• s: segment.
• cir: circle.
• str: string.
• flg: flag.

An angle measure is a number expression followed by `:' (degrees) or `<' (radians). A string is any text contained in one line and enclosed in double quotes.

We'll use square brackets to indicate optional parameters.

2. Handling numbers and number valued functions

To compute usual arithmetic operations, use the symbols : `(', `)', `+', `-', `*', `/', `^' (for exponentiation).

Here are all the number valued functions:

• sqrt(x)
• pi
• sin(x)
• cos(x)
• tan(x)
• asin(x)
• acos(x)
• atan(x)
• deg(x)
• rad(x)
• abscissa(A) or abscissa(u)
• ordinate(A) or ordinate(u)
• distance(A, B) or distance(A, l)
• length(u) or length(s)
• radius(cir)
• heigth(A, B, C)
  Distance between point B and line (AC).
• angle(u) or angle(l) or angle(s)
  Argument of the object, i.e. measure of the angle between the object and the horizontal direction.
• angle(u, v)
  Measure of the angle between vector u and vector v.
• angle(A, B, C)
  Measure of the angle under ABC.

3. Handling vectors

To compute usual vector operations, use the symbols : `(', `)', `+', `-', `*' (multiply by a number), `/' (divide by a number). To compute the scalar product of two vectors, use the symbol : `*'.

Here are all the vector valued functions:

• vector(x, y)
  Vector of abscissa x and ordinate y.
• vector(x, ang)
  Vector of length x and argument ang.
• vector(A, B)
  Vector from point A to point B.
• vector(l)
  Unit length vector with same direction than line l.
• vector(s)
  Vector from begin of segment s to end of s.
• rotation(u, ang)
  

4. Point valued functions

Here are all the point valued functions, except See section 8. Geometrical transformation functions.

• point(x, y)
  Point of abscissa x and ordinate y.
• point(x, ang)
  Point of radial coordinate x and argument ang.
• point(l, x)
  Point of abscissa x on the graduated line l.
• point(s, x)
  Point of abscissa x on the graduated line containing s (the origin is set to the begin of s).
• point(cir, ang)
  Point of argument ang on circle cir.
• projection(A, l1 [, l2])
  Projection of point A on line l1 in direction of line l2 (default: in perpendicular direction of line l1).
• barycenter(A [, x], B [, y])
  Barycenter of A and B with coefficient x and y (default: 1 and 1). Similar syntax for 3 or 4 points.
• intersection(l1, l2)
  Intersection point of lines l1 and l2.
• abscissa(l, x)
  Point of abscissa x on line l.
• ordinate(l, y)
  Point of ordinate y on line l.
• midpoint(s)
• begin(s)
• end(s)
• center(cir)
• orthocenter(A, B, C)
  Orthocenter of triangle ABC.

5. Line valued functions

Here are all the line valued functions, except See section 8. Geometrical transformation functions.

• line(A, B)
• line(A, u)
• line(s)
• line(A, ang)
  Line containing point A with argument ang.
• line(cir, ang)
  Tangent line of circle cir containing point of argument ang.
• parallel(l, A) or parallel(s, A)
• perpendicular(l, A) or perpendicular(s, A)
• bissector(s)
  Perpendicular bissector of segment s.
• bissector(A, B, C)
  Bissector of angle under ABC.
• altitude(A, B, C)
  Altitude of triangle ABC containing point A.
• median(s)

6. Segment valued functions

Here are all the segment valued functions, except See section 8. Geometrical transformation functions.

• segment(A, B)
• segment(A, u)
• segment(A, x, ang)
  Segment of length x and argument ang, beginning with A.
• segment(cir, ang)
  Segment from center of circle cir to point of argument ang.

7. Circle valued functions

Here are all the circle valued functions, except See section 8. Geometrical transformation functions.

• circle(A, B)
  Circle of diameter [AB]
• circle(A, B, C)
  Circumcircle of triangle ABC.
• circle(A, x)
  Circle of center A and radius x.
• incircle(A, B, C)

8. Geometrical transformation functions

We'll note obj an object which can be a point, a line, a segment or a circle. Here are all the transformation functions:

• translation(obj, u)
• reflection(obj, l)
• rotation(obj, A [, ang])
  Rotation of center A and angle ang (default: 180 degrees).
• homothecy(obj, A, x)
  Homothecy, i.e. expansion or dilation, of center A and ratio x.

9. Defining triangles

A triangle assignment is a list of 3 variable names followed by the word triangle, right, isosceles or equilateral and some optional parameters. If the first variable is already defined as a point, the triangle will be constructed from this point. If not, the point will be set to origin.

Here are all the ways to define a triangle:

• A B C triangle[(x [ , ang])]
  Defines a scalenes triangle with AB = x (default: 6). The triangle is an optimal scalenes triangle.
• A B C triangle(x, y, z [ , ang])
  Defines a scalenes triangle with AB = x, BC = y and CA = z.
• A B C triangle(x, ang1, ang2 [ , ang3])]
  Defines a scalenes triangle with AB = x, measure of the angle under BAC equals ang1, measure of the angle under CBA equals ang2.
• A B C right[(x, y [ , ang])]
  Defines a right triangle (right angle in A) with AB = x and AC = y (default: 6 and 4.5).
• A B C right(x, ang1 [ , ang2])
  Defines a right triangle (right angle in B) with AB = x and measure of the angle under BAC equals ang1.
• A B C isosceles[(x, ang1 [ , ang2])]
  Defines an isosceles triangle with AB = x and measure of the angle under BAC or CBA equals ang1 (default: 6 and 39 degrees).
• A B C isosceles(x, y [ , ang])
  Defines an isosceles triangle with AB = x and AC = CB = y.
• A B C equilateral[(x [ , ang])]
  Defines an equilateral triangle of side length x (default: 6).

Note: The last optional parameter is the argument of segment [AB] (default: 0 degrees).

10. Defining polygons

A quadrilateral assignment is a list of 4 variable names followed by the word parallelogram, rectangle or square and some optional parameters. If the first variable is already defined as a point, the quadrilateral will be constructed from this point. If not, the point will be set to origin.

Here are all the ways to define a quadrilateral:

• A B C D parallelogram[(x, y, ang1 [ , ang2])]
  Defines a parallelogram with AB = x, AD = y and angle under BAD equals ang1 (default: 5, 4 and 75 degrees).
• A B C D parallelogram(u, v [ , ang])
  Defines a parallelogram with vector from A to B equals =u and vector from A to D equals v.
• A B C D rectangle[(x, y [ , ang])]
  Defines a rectangle with AB = x and AD = y (default: 6 and 6 / `golden ratio').
• A B C D square[(x [ , ang])]
  Defines a square with AB = x (default: 4).

Note: The last optional parameter is the argument of segment [AB] (default: 0 degrees).

A pentagon assignment is the following command:

• A B C D E pentagon(F, x, ang)

An hexagon assignment is the following command:

• A B C D E hexagon(G, x, ang)

11. Intersection points

• A B intersection(l, cir)
• A B intersection(cir1, cir2)

12. Interactive assignements

• x interactive(y, z, [ x1 , x2 , ] str, flg)

13. General graphical commands

The purpose of general graphical commands is mainly to set some general parameters.
Here are all of them:

• frame(x1, y1, x2, y2 [ , z])
  Sets the visible frame. The lower-left corner has coordinates (x1, y1) and the upper-right corner (x2, y2). With euklides, the optional parameter set the unit length to z cm (default: 1). With xeuklides, this parameter has no effect, because the figure is always drawn using the largest scale such that it fits into the drawing area. If no frame is specified it is set to default frame: (-2, -2, 8, 6).
• color(flg)
  Sets color to flg. Permitted values of flg are black (default), darkgray, gray, lightgray, white, red, green, blue, cyan, magenta and yellow.
• color str
  With euklides, sets color to str (it has to be a valid pstricks color). With xeuklides, it has no effect.
• tricks str
  With euklides, adds an initial `\' to str and puts it verbatim to the output. It has to be valid TeX code. You can use it to define your own colors, draw a curve, etc. With xeuklides, it has no effect.
• font str
  With xeuklides, changes current font. String str has to be a valid XLFD. With euklides, it has no effect.

14. Drawing commands

Here are all the drawing commands.

• draw(A [ , flg])
  Draws point A with a shape corresponding to flg. Permitted values of flg are dot, box, cross and plus (default: dot).
• draw(u, A [ , flg])
  Draws vector u from point A with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(l [ , flg1 [ , flg2]])
  Draws line l with an aspect corresponding to flg1 and flg2. Permitted values of flg1 are full, dashed and dotted (default: full). Permitted values of flg2 are entire, halfline and backhalfline (default: entire).
• draw(s [ , flg1 [ , flg2]])
  Draws segment s with an aspect corresponding to flg1 and flg2. Permitted values of flg1 are full, dashed and dotted (default: full). Permitted values of flg2 are noarrow, arrow, backarrow and doublearrow (default: noarrow).
• draw(cir [ , flg])
  Draws circle cir with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(cir, ang1, ang2, [ , flg1 [ , flg2]])
  Draws an arc of circle cir from point of argument ang1 to point of argument ang2, with an aspect corresponding to flg1 and flg2. Permitted values of flg1 are full, dashed and dotted (default: full). Permitted values of flg2 are noarrow, arrow, backarrow and doublearrow (default: noarrow).
• draw(A, B, C, [ , flg])
  Draws a triangle with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(A, B, C, D [ , flg])
  Draws a quadrilateral with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(A, B, C, D, E, [ , flg])
  Draws a pentagon with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(A, B, C, D, E, F [ , flg])
  Draws an hexagon with an aspect corresponding to flg. Permitted values of flg are full, dashed and dotted (default: full).
• draw(str, A, [ x, ] ang)
  Prints the text contained in str at a distance x (default: .3) and with argument ang from point A.
• draw(str, s, [ x, ] ang)
  Prints the text contained in str at a distance x (default: .3) and with argument ang from the midpoint of segment s.
• draw(x, [str, ] A, [ y, ] ang)
  Prints the value of x optionaly formated with str (using the C-language syntax) at a distance y (default: .3) and with argument ang from point A.
• draw(x, [str, ] s, [ y, ] ang)
  Prints the value of x optionaly formated with str (using the C-language syntax) at a distance y (default: .3) and with argument ang from the midpoint of segment s.
• draw(x1, x2, str, A, [ y, ] ang)
  Prints the values of x1 and x2 formated with str (using the C-language syntax) at a distance y (default: .3) and with argument ang from point A.
• draw(x1, x2, str, s, [ y, ] ang)
  Prints the values of x1 and x2 formated with str (using the C-language syntax) at a distance y (default: .3) and with argument ang from the midpoint of segment s.
• mark(s [ , flg])
  Marks the segment s with a symbol corresponding to flg. Permitted values of flg are simple, double, triple and cross (default: simple).
• mark(A, B, C [ , flg])
  Marks the angle under ABC with a symbol corresponding to flg. Permitted values of flg are simple, double, triple, dash, arrow, backarrow and right (default: simple).

15. Samples

An isosceles triangle with some of its elementary properties illustrated.

A B C isosceles ; draw(A, B, C) H = projection(C, line(A, B)) ; draw(segment(C, H), dashed) ; mark(B, H, C, right) mark(segment(A, C), cross) ; mark(segment(C, B), cross) mark(B, A, C, dash) ; mark(C, B, A, dash)

A figure illustrating a property of the incircle in a triangle.

A B C triangle ; draw(A, B, C) ; draw(incircle(A, B, C)) draw(bissector(B, A, C), dotted) draw(bissector(A, B, C), dotted) draw(bissector(B, C, A), dotted)

Addition of two vectors.

A B C D parallelogram draw(segment(A, B), full, arrow) ; draw(segment(A, C), full, arrow) ; draw(segment(A, D), full, arrow) draw(segment(B, C), dotted) ; draw(segment(D, C), dotted)

An angle property of parallelograms.

A B C D parallelogram(5, 4, 105:) ; draw(A, B, C, D) mark(B, A, D) ; mark(D, C, B) mark(C, B, A, double) ; mark(A, D, C, double)

An hexagon and its diagonals.

A B C D E F hexagon(point(3,2), 3, 0:) ; draw(A, B, C, D, E, F) draw(segment(A, D), dotted) draw(segment(B, E), dotted) draw(segment(C, F), dotted)

Contact

Euklides has been created by Christian Obrecht.

Please send bug reports and comments at obrecht@mail.chez.com.

Concept index

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Command index

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Table of Contents

Overview of the Euklides system
1. Design of the language
2. Handling numbers and number valued functions
3. Handling vectors
4. Point valued functions
5. Line valued functions
6. Segment valued functions
7. Circle valued functions
8. Geometrical transformation functions
9. Defining triangles
10. Defining polygons
11. Intersection points
12. Interactive assignements
13. General graphical commands
14. Drawing commands
15. Samples
Contact
Concept index
Command index

Short Table of Contents

Overview of the Euklides system
1. Design of the language
2. Handling numbers and number valued functions
3. Handling vectors
4. Point valued functions
5. Line valued functions
6. Segment valued functions
7. Circle valued functions
8. Geometrical transformation functions
9. Defining triangles
10. Defining polygons
11. Intersection points
12. Interactive assignements
13. General graphical commands
14. Drawing commands
15. Samples
Contact
Concept index
Command index

About this document

• 1. Section One
• 1.1 Subsection One-One
• ...
• 1.2 Subsection One-Two
• 1.2.1 Subsubsection One-Two-One
• 1.2.2 Subsubsection One-Two-Two
• 1.2.3 Subsubsection One-Two-Three     <== Current Position
• 1.2.4 Subsubsection One-Two-Four
• 1.3 Subsection One-Three
• ...
• 1.4 Subsection One-Four