Getting Started/Brief Walkthrough
Measuring and Calculating
Editing and Reconstraining
Views and Image Exporting
Getting Started/Brief Walkthrough
After doing the following, you should be familiar with the most basic usage of
KSEG and can start playing around on your own. For advanced features,
read this entire document.
Of course, this is not the most efficient way to construct a
perpendicular bisector in KSEG (although it's the
compass-and-straightedge way), but the point is to show how
construction and selection work.
- Start KSEG. You should see a blank white canvas taking up most
of the sketch window (surrounded by toolbars, scroll bars, menu, and
- Right click in the middle of the white canvas to construct point A.
- Hold SHIFT as you right click again in a different spot on the
canvas to construct point B. Since you are holding SHIFT, point A is
not deselected, so after this step you should have two selected points.
- Choose "New/Circle By Center And Point" from the menu. This
should construct a circle centered at point A and passing through
point B. The circle will appear selected and points A and B will be
- Select point B by left clicking on it (this also deselects the
circle). Then hold SHIFT and left click on point A to select it.
- Choose "New/Circle By Center And Point" from the menu again. Now
this should construct a circle centered at B (because you selected it
first) passing through A.
- Hold SHIFT and select the first circle you constructed by
clicking on it. You should now have both circles selected.
- Choose "New/Intersection Points" from the menu. Two points at
the intersections of the two circles should appear, both of them
- Choose "New/Line" from the menu. This will construct the
perpendicular bisector of points A and B.
- Now drag point A with the left mouse button. This should give
you an idea of what KSEG does.
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KSEG is a tool designed to let you easily visualize dynamic properties
of compass-and-straightedge constructions and to make geometric
exploration as fast and easy as possible. Almost every time I sit
down to play with KSEG, I discover a geometric property that I didn't
With KSEG, you create a dynamic construction by first placing points
on the screen and then using them to construct other objects, such as
lines or circles. Then you can drag the points you created and watch
the entire construction respond. There is much more to KSEG, of
course, but this is the basic idea. I also often use it to generate
high-quality geometric diagrams for use with LaTeX. I find that the
easiest way to do this is to print from KSEG to a .ps file, convert
it to EPS, annotate it in XFIG, and export to the PS/LaTeX format.
A terminology problem: I use the term "Construction" to
refer to the macro-like document type in KSEG as well as to just a
bunch of geometric objects that may depend on each other. It is
usually clear from the context which I mean, but in this document I
will capitalize "Construction" to refer to the document type and will
use the lower case "construction" to refer to a bunch of geometric
Points, lines, measurements, calculations, loci, etc. are all
"geometric objects". A geometric object may have parent geometric
objects on which it depends. For example if you construct two points
and a line through them, then the points are parents of the line and
the line is a child of the points. When you drag a free point, all of
its descendants are recalculated. When you drag an object that has
parents (this is called reverse dragging), a complicated algorithm
determines which parents to move in order to get the desired motion of
the object you are dragging.
A geometric object may be "non-existent" in the geometric sense, such
as the intersection point of circles that do not intersect. If the
circles are later moved closer together, the points begins to exist
again. Children of non-existent objects are also non-existent (with
the exception of the locus, for details see locus documentation
below). So if you are dragging around a point and all of a sudden
most of your construction disappears because two line segments ceased
to intersect (the objects that disappeared are the descendants of the
intersection point), don't worry, it will come back as soon as they
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In order to construct new objects, you usually need to select existing
ones. Here are your options for doing this:
When you click on a spot where there is more than one object, things
are more complicated (for KSEG, that is--for you it should seem very
intuitive). Objects have a selection "precedence": points first,
curves and text and loci second, and filled objects last. If objects
with higher precedence are under the mouse then objects with lower
precedence are ignored. Within a single precedence, the algorithm is
such that clicking multiple times cycles through some selection
possibilities. The status bar is always a good place to look to find
out what will happen if you click.
- Left clicking on an empty spot deselects all currently selected
- Left clicking on an unselected object selects that object
and deselects all the other ones.
- Pressing the left mouse button on an empty spot, dragging, and
releasing will select all objects that intersect the dragged rectangle
and deselect all others.
- Holding down shift and left clicking on an object will select the
object if it is not already selected and deselect it otherwise. Other
objects that may have been selected remain selected.
- Holding down shift, pressing the left mouse button on an empty
spot, dragging, and releasing will add all of the objects that
intersect the dragged rectangle to the selection.
- In a Construction, you also have the option of selecting objects
through the Construction list. You can click or drag or use shift.
Note that unlike the other methods, this allows you to select
invisible and non-existent objects.
A more powerful way of selecting multiple objects is to use selection
groups. A selection group is a bunch of objects that you tell KSEG to
group together to make it easy to do selection operations with them.
Going to "Edit/Selection Groups..." brings up the selection groups
manager window (associated to a document). When you open it, it
displays in a list box all selection groups you have for the document
in addition to the built-in selection groups. The built-in selection
groups are marked with an asterisk--there is one for every type of
object that groups all the objects of that type.
When you select a selection group (or several) in the listbox of the
selection group manager and click the Select button, all objects of
that group are selected and all other objects are deselected. Holding
shift while clicking the Select button will select the objects in that
group without deselecting other objects that may be currently
selected. Clicking the Deselect button will deselect only the objects
in the group. When you have at least one selected object, the New
Group button is enabled. Click it to create a new group consisting of
the selected objects. You cannot delete built-in groups.
The Affect Invisible Items checkbox only affects the built-in
selection groups. When it is unchecked (default), "All Points" refers
to only the non-hidden points; when it is checked, "All Points" refers
to the hidden ones as well. Groups that are not built-in always
select/deselect all objects in them, including hidden ones.
A (non-built-in) group may become empty if all its items are deleted
(directly or through undo). Such groups are marked with a minus sign
but are not automatically deleted (since undo/redo may bring objects
in those groups back to life). However, empty groups are not saved
with the document.
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When new objects are constructed, they are immediately selected and,
with the exception of points constructed by right-clicking with shift
held down, all other objects are deselected.
There are also two other kinds of points--endpoints and center points.
These are mostly useful for Constructions, where you may have as a
Given a segment but not its endpoints or a circle but not its center.
To construct the endpoint(s) of a segment, ray, or an arc, select it
and go to "New/Endpoint(s)". To construct the center point of a
circle or an arc, select it and go to "New/Center Point".
- To construct a free point, right click on an empty area of the
- To construct a point constrained to a curve (line, segment, ray,
circle, arc), right click on the curve.
- To construct a point constrained to an intersection of two
curves, right click on the intersection. Alternatively, select the
two curves and choose "New/Intersection point(s)" from the menu. When
the two curves have two intersection points (at least potentially) the
menu option will construct both of them, while right-clicking will
only construct one.
- To construct the midpoint(s) of one or more line segments, select
them and choose "New/Midpoint(s)".
There is only one way two make a line segment--select two or more
points and choose "New/Segment(s)". If two points are selected, one
segment will be constructed. If n > 2 points are selected
then n segments will be constructed.
- You can construct a ray from one point through another by
selecting the two points (in order) and going to "New/Ray". Select
more than two points to construct multiple rays.
- To construct the bisector of an angle, select three points
defining the angle and go to "New/Angle Bisector". The bisector
always bisects the acute part of the angle.
- To construct a line, select two or more points and go to "New/Line(s)".
- To construct a line parallel to a given one through a point,
select the point and the line (you can use a segment or a ray as well)
and go to "New/Parallel Line". You can select multiple points or multiple
straight objects (but not both).
- To construct a perpendicular line, select as for the parallel line, but
choose "New/Perpendicular Line(s)" from the menu instead.
- To construct a circle with a given center through a given point, select
the center and then the point through which the circle should pass and go
to "New/Circle By Center And Point".
- To construct a circle with a given center and a given radius, select
the center point and a line segment whose length will be the radius and
choose "New/Circle By Center And Radius".
An arc can only be constructed by selecting three points and going to
"New/Arc By Three Points".
A locus is a geometric object that represents the path of another
geometric object (the "driven" object) as a point (the "driver") moves
along a curve. To construct a locus, select a point on a curve and a
descendant of that point. The driven object can be a point, a curve,
or even another locus (but this can get really slow). A locus could
exist even if the driven object is non-existent, since the driven
object may exist when the driver is in a different position.
For example, let L be a line, let
A be a point on L, let B be a point not on
L, and let K be the line through A perpendicular
to AB. Then if K is the driven object and A is
the driver, the envelope of the locus traced by K is the
parabola with B at its focus that is tangent to L.
Since a locus is an infinite set of geometric objects, KSEG cannot
compute it exactly. Instead, it approximates the locus by sampling.
You can change the number of samples KSEG performs for a particular
locus by either double-clicking on that locus or selecting it and
choosing "Edit/Change Number Of Samples..." A special case is when
the driven object is a point. In that case, KSEG automatically
connects sampled points and uses an adaptive algorithm to try to make
the locus smooth and accurate quickly. In that case, the number of
samples you specify is only an upper bound on the number of samples
actually performed--the algorithm may decide to stop sampling if it
thinks the locus is good enough. This adaptive algorithm generally
results in much better looking loci compared to those generated by
programs similar to KSEG.
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- To construct a polygon interior, select three or more points and
- To construct an arc sector, select an arc and choose "New/Arc
Sector". For an arc segment, select an arc and choose "New/Arc Segment".
- To construct a circle interior, select a circle and choose
KSEG allows you to apply some affine transforms to geometric
objects. The general procedure is to "choose" some objects to define
the transform using the "Transform/Choose *" menu items. Then
you select the objects to transform, choose the appropriate action
from the transform menu, and transformed copies of the selected objects
will be created. Here are the details:
The currently chosen groups of objects are marked with yellow to let
you know what the transforms will do. If you are done with the
transforms and want to get rid of the yellow markings (for aesthetic
or screenshot purposes--they don't appear when printing), go to
- To choose a vector, select two points and go to "Transform/Choose
Vector". The vector will be from the first selected point to the
- To choose a mirror, select a line, a segment, or a ray and go to
- To choose a center, select a point and go to "Transform/Choose
- To choose an angle, select either three points or a single
measurement or calculation and go to "Transform/Choose Angle".
- To choose a ratio, select either two line segments or a single
measurement or calculation and go to "Transform/Choose Ratio".
Being able to transform objects with a measurement or calculation
chosen for the ratio or the angle easily allows you to create
constructions impossible with a compass and a straightedge. You may
want to be careful with this, but it can also be fun--the example
coord.seg file shows how it can make KSEG graph parametric curves as
- To translate objects you need to have a vector chosen.
- To reflect objects, you need to have a mirror chosen.
- To scale objects, you need a center and a ratio chosen.
- To rotate objects, you need a center and an angle chosen.
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Measuring and Calculating
You can measure various geometric quantities with KSEG and perform
calculations with the measurements. All distance units are in pixels
and angles are in degrees. Here's what you can measure:
You can combine several measurements in a calculation. To create a
calculation, either choose "Measure/Calculate..." from the menu or
double-click on an existing measurement. The menu option creates an
"empty" calculation if nothing was selected and but if an measurement
or calculation was selected, it creates a calculation that references
the selected one. Double-clicking creates a calculation that
references the measurement you double-clicked on. In either case, the
calculation editor window appears in which you can enter the formula
you want calculated. If you double-click on an existing calculation,
a calculation editor window will come up allowing you to edit that
- To measure the distance between a point and either a curve or another
point, select the two geometric objects and go to "Measure/Distance".
- To measure the length of a line segment, select the line segment
and go to "Measure/Length".
- To measure the circumference of a circle, select the circle and
go to "Measure/Circumference".
- To measure the arc length of an arc, select the arc and go to
- To measure the radius of an arc or a circle, select the object
and go to "Measure/Radius".
- To measure the angle defined by three points, select the three
points and go to "Measure/Angle".
- To measure the ratio between the lengths of two line segments,
select them and go to "Measure/Ratio".
- To measure the slope of a line segment, a ray, or a line, select
the object and go to "Measure/Slope".
- To measure the area of a filled object, select the polygon, the
arc sector, the arc segment, or the circle interior and go to
"Measure/Area". If you have a "bad" polygon (sides don't form a
simple closed curve), its measured "area" will not be equal to the
area on the screen that it appears to have.
The undo stack in the calculation editor is completely separate from
the undo stack in the rest of KSEG and is lost as soon as the
calculation editor window is closed.
Within the calculation editor window, you can enter numbers, usual
arithmetic operations, select roots, powers, and fractions from the
toolbar (or use shortcut keys CTRL+2, CTRL+6, CTRL+/). You can also
use the following functions:
sqrt, log, exp, floor, ceil, fabs, sin, cos, tan, sinh, cosh,
tanh, asin, acos, atan, asinh, acosh, atanh
Note that the above functions work with radians, in contrast to the
measurements/transforms that are in degrees. I know this sucks, but
the only reasonable way to fix it is to add unit support.
Finally, you can reference other measurements and calculations by
left-clicking on them in the main KSEG window while the calculation
editor is open. You will not be allowed to reference calculations
that reference the one you are editing.
I must say that this is the part of KSEG I am least happy with. There
is no support for any reasonable units and the calculations are kind
of awkward. Finally, the formula code is ugly and slow (and somewhat
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Editing and Reconstraining
Appearance Modification and Labels
You can specify how you want a particular geometric object to be drawn
by selecting the object (or several of them) and going to
"Edit/Color", "Edit/Linestyle", "Edit/Pointstyle", or "Edit/Font", as
appropriate. Going to these menus with no objects selected will
modify the default appearance of subsequently constructed objects.
Even though colors, fonts, etc. are not available in a toolbar, you
can click on the dashed line at the top of one of these menus to "tear
it off" and get an effect similar to a toolbar.
Points and the various curves have labels that you can display (they
are not shown by default) by selecting the object(s) and going to
"Edit/Show Label(s)". They can be modified by selecting the object
and going to "Edit/Change Label" or double-clicking on the label if
it is shown. You can even give labels subscripts and superscripts
as well as other stuff if you can figure out how :-) You can drag
a label to move it relative to the object it refers to.
Nothing is more frustrating than spending half an hour on an elaborate
construction and then discovering that you made a small mistake in the
beginning. KSEG's reconstraining mechanism will often let you easily
To reconstrain a point, hold down CTRL, left-drag the point, and drop
it onto the object(s) you would like to reconstrain it to.
example, suppose you construct a triangle ABC then do a long
construction of a yet undiscovered center, and then decide to
investigate its behavior when ABC is isosceles. To make
AB=BC, construct the circle with center B through
A. Then hold CTRL and drag point C onto the circle.
After you release the mouse button, C will be constrained to
lie on the circle and so ABC will be isosceles. If you decide
to make ABC arbitrary again, hold CTRL, and drag C of
the circle and drop it to an empty spot.
All of this is of course undoable with CTRL+Z ("Edit/Undo"). The
status bar will do its best to keep you informed of what will happen
if you drop the point.
- Holding CTRL and starting to drag a point immediately makes the
- Dropping the point on an empty spot makes the point stay free.
- Dropping the point on a curve constrains the point to lie
on that curve.
- Dropping the point on the intersection of two curves constrains
the point to lie on that intersection.
- Dropping the point on another point will merge the two points.
Internally, the point you dragged will disappear and all of its
children will be transferred to the point you dragged onto.
Sometimes KSEG will not let you do the reconstraining you want. This
is for good reason (although there may be bugs): you cannot
reconstrain a point to one of its descendants. You also cannot
reconstrain a point if doing so will break the dependence of a driven
point in a locus on its driver (you also can never reconstrain a point
that is the driver of some locus). Finally, there are complicated
rules in Constructions that try to ensure that you don't mess the
Construction up with a reconstraint. For example, you will not be
able to reconstrain a "Given" point to a "Final" curve. If you find
that KSEG allows you to perform a reconstraint that makes the
Construction invalid, please let me know.
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Views and Image Exporting
View Panning and Zooming
In order to get a better look at your construction, KSEG allows you to
zoom in or out and move it around. To zoom, go to "View/Zoom". The
mouse cursor should change into a magnifying glass (when over the
sketch area). Press the left mouse button and drag up and down to
zoom in or out. The zoom is around the point where the mouse button
was pressed. After the mouse button is released, the mouse cursor
will change back to the normal arrow. The current magnification is
always shown in the status bar. Notice that regardless of
magnification, things like points, line thicknesses, and fonts do not
change size. Pan works in a similar way to zoom--go to "View/Pan",
press the left mouse button and drag around. Alternatively, you can
pan without going to the menu--just press the middle mouse button and
If you go to "View/Zoom To Fit", KSEG will automatically pan and zoom
the image so that it just fits in the view. If you choose
"View/Original Zoom(100%)", KSEG will set the magnification to 100%
and will pan the image back to the origin.
To create another view (in another window) of the current document, go
to "View/New View". A new window will appear. Panning and zooming in
one view will not affect any other views, but modifying the document
or selecting objects in it is reflected in all views of that document.
So if you drag a point around in one view, you will see it responding
in all views. You can use views to, for example work on a small part
of a sketch (or make fine motions in a zoomed-in view) while
simultaneously seeing the effects on the entire sketch in a different
You can export the contents of the current view to an image file (in
several popular formats). To do this, go to "File/Export To
Image...". A dialog appears that allows you to choose the file format
and whether you want the image to be "High Quality". Click OK and
then choose the filename in the dialog that appears. The dimensions
of the image will be the dimensions of the current view.
If you select "High Quality Image", KSEG will take longer to export,
but the image will be antialiased and drawn with subpixel precision.
This option is great for making web graphics, but it might look
horrible if you print it (worse than the low quality version). Also,
be sure not to use the high quality setting if you are exporting to a
black-and-white image (but grayscale is fine)--it will look ugly.
Using low quality mode is "safer" since the image exported is exactly
what is on the screen.
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Constructions are probably the most powerful feature of KSEG. A
Construction is like a normal KSEG sketch, except that it can be
played back to another KSEG sketch (or Construction). First,
a simple example:
Circumcircle Example Walkthrough
This part will walk you through the construction of the
circumcircle.sec example Construction. Along the way, I will explain
what each step means. This Construction will, given three points,
construct the circle that goes through them.
- Create a new Construction by going to "File/New Construction".
- Construct three points A, B, and C. These points will represent
the ones given to the Construction.
- Select A, B, and C and make them "Given" by going to
"Construction/Make Given". This lets KSEG know that these points will
be given to the Construction when it is played back.
- Construct line segments AB and BC and their midpoints, D, and E.
- Construct the line through D perpendicular to AB and the line
through E perpendicular to BC. Construct their intersection F.
This is the circumcenter of triangle ABC.
- Now construct the circle with center F that goes through the
point A. This is the desired circumcircle.
- Now select all the objects you constructed except for the
circumcircle and the three Given points and go to "Edit/Hide Objects"
to make them hidden. We do this is because when we play back the
Construction, we don't want the intermediate construction objects
cluttering up the screen--we just want the circumcircle.
- That's it! Test the Construction by creating a new sketch (with
"File/New Sketch"), constructing three points in the sketch, selecting
them and going to "Play/Untitled" (assuming you didn't save the
Construction). A circumcircle should magically appear!
Most of the power of Constructions comes from their ability to be
recursive. Hence, non-recursive Constructions are very simple--the
above circumcircle example covers almost everything there is to know
about them. Here are a few details (they apply to recursive
Constructions as well):
- Objects with parents can be marked as Given. In that case, it
often makes sense not to construct the parents, if they are not useful
for anything else. Such objects are marked "(not constructed)" in the
Technically, this is the rule: Imagine the parent-child
relationships as a directed graph on the objects. If every path (in
the directed graph) from object A to every object with no
children has at least one Given object, then A is not
- You will not be allowed to mark as Given objects that are
necessary for the definition of a locus. For example, you can't mark
the driver point as Given because KSEG can't guarantee that it is
constrained to a curve when the Construction is played back.
- It is possible for the objects selected to be matched to Given
objects whose types are different. For example, if in a Construction,
you create segment S, mark it as Given, and construct a point
on it, you may select (in a different document) an arc, circle, ray,
line, or segment, play this Construction, and a point will appear on
the object you selected. However, if in the Construction you
construct a perpendicular line to S through the point, you will
not be able to play that Construction on a circle or an arc.
The algorithm that matches selected objects to the Given objects
is not trivial (but it's not very good theoretically either). The
first thing that is tried is matching the selected order to the Given
order. If that doesn't work, a search is tried (it really should do a
bipartite matching, but it's more work to implement and the value is
questionable). Sometimes it may not be able to match the selected
objects to the Givens even when such a match is possible (and the item
in the Play menu will be disabled). In that case, just select the
objects in a different order, closer to the Given order.
- Free points (as well as measurements and calculations) that are
not marked as Given will be placed randomly. Points constrained to
curves will be placed at random positions on the curve.
- Quick Play is a very handy feature for accessing a library of
Constructions you commonly use. Select the directory that stores
your favorite Constructions with "Play/Quick Play/Change
Directory...", then select some objects in your current document that
correspond to the Givens of one of the Constructions in that directory
and go to "Play/Quick Play/ConstructionFilename.sec" to play it without
actually loading the Construction. For this to work, KSEG scans the
Constructions in the directory and enables those whose Givens match
the currently selected objects.
- Copy As Construction allows you to take a sketch and change it
into a Construction document. This is useful if you were playing
around with a sketch and found something you'd like to apply to other
sketches--go to "File/Copy As Construction" and a new Construction
appears containing the same geometric objects as the sketch. You can
then delete objects you don't need, mark some Given objects and apply
the Construction to other documents.
The coolest way to use Constructions is recursively. This allows you
to construct fractals, as well as just repeat Constructions multiple
times. For a simple example, see tractix.sec (use a short segment and
recursion depth of like 30). For a more complicated one (with
multiple recursion branches), see dragon.sec, which constructs a
A Construction becomes recursive when you insert loop "objects" into
it (they aren't really geometric objects, they just pretend to be).
To insert a loop, select some objects that can match to the Givens
and go to "Construction/Recurse". A loop tells KSEG that after all
other steps in the Construction have been performed, it should repeat
the Construction recursively, starting with the objects selected at
the time the loop is specified. Of course, KSEG doesn't do this
forever--when you play back a recursive Construction, KSEG asks you
for the maximum recursion depth. Be careful--if you have more than
one loop in the Construction, the number of objects KSEG has to
construct is exponential in the maximum recursion depth--in other
words, keep the recursion depth small. On my old 350mhz system, KSEG
can handle about 100,000 objects comfortably--this should be enough
for anything I can think of.
Remember that as soon as you insert a loop into a Construction, the
set of Given objects becomes fixed (you will not be able to mark new
objects as Given, make currently Given objects normal, deleting a
Given object deletes all the loops, etc.) This is to prevent the
loops from becoming invalid. To change the set of Given objects,
delete the loops first.
You can also make the objects in a recursive Construction "Final" or
"Initial". Final objects are straightforward: they are only
constructed at the maximum recursion depth. This is useful for many
things, particularly for fractals that only show the highest level
of detail--see dragon.sec or sierp2.sec. Children of Final objects
can also be constructed only at the maximum recursion depth--they are
marked "(final)" in the Construction list.
Initial objects are complicated--it's an "advanced" feature. It's not
all that powerful, but it has enough weird effects for it to be really
hard to figure out sometimes. An Initial object is only constructed
once (at the top level recursion depth). Any object all of whose
parents are Initial (or implicitly Initial) becomes implicitly Initial
and is marked with "(initial)" in the Construction list. For a good
illustration, see sierp.sec: first play it back normally (notice that
the largest sides of the triangle are constructed only once) and then
construct a point P on the Initial segment s_9 and construct line
segment PC (from P to the Given point C). Play it back again and
observe the effect. Initial objects can be used to conserve memory
(as in koch.sec) or for their effects.
In recursive Constructions, when dealing with Final/Given/Initial
objects, KSEG will do its best to prevent invalid or meaningless
configurations, such as Initial or Given objects having Final objects
for parents. It may not be obvious what it's doing, but it is usually
correct (if you find a case where it isn't, email me please, with the
valid Construction and a description of the action that makes it
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