# KSEG Help

## Quick Jump:

Getting Started/Brief Walkthrough
Introduction
Selecting
Constructing
Transforms
Measuring and Calculating
Editing and Reconstraining
Views and Image Exporting
Constructions

## 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.
1. Start KSEG. You should see a blank white canvas taking up most of the sketch window (surrounded by toolbars, scroll bars, menu, and status bar).
2. Right click in the middle of the white canvas to construct point A.
3. 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.
4. 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 deselected.
5. 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.
6. 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.
7. Hold SHIFT and select the first circle you constructed by clicking on it. You should now have both circles selected.
8. Choose "New/Intersection Points" from the menu. Two points at the intersections of the two circles should appear, both of them selected.
9. Choose "New/Line" from the menu. This will construct the perpendicular bisector of points A and B.
10. Now drag point A with the left mouse button. This should give you an idea of what KSEG does.
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.

## Introduction

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 know before.

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 objects.

### Geometric Objects

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 which 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 intersect again.

## Selecting

In order to construct new objects, you usually need to select existing ones. Here are your options for doing this:
• Left clicking on an empty spot deselects all currently selected objects.
• 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.
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.

### Selection Groups

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 which 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.

## Constructing

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.

### Points

• To construct a free point, right click on an empty area of the KSEG sheet.
• 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 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".

### Segments

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.

### Rays

• 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.

### Lines

• 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.

### Circles

• 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".

### Arcs

An arc can only be constructed by selecting three points and going to "New/Arc By Three Points".

### Loci

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.

### Filled Objects

• To construct a polygon interior, select three or more points and choose "New/Polygon".
• 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 "New/Circle Interior".

## Transforms

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:

Choosing:

• To choose a vector, select two points and go to "Transform/Choose Vector". The vector will be from the first selected point to the second.
• To choose a mirror, select a line, a segment, or a ray and go to "Transform/Choose Mirror".
• To choose a center, select a point and go to "Transform/Choose Center".
• 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".
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 "Transform/Clear Chosen".
• 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.
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 loci.

## 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:
• 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 "Measure/Arc Length".
• 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.
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 which references the selected one. Double-clicking creates a calculation which 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 calculation.

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/tranforms which 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 which 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 buggy).

## 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.

### Reconstraining

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 fix it. 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.

For 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.

So:

• Holding CTRL and starting to drag a point immediately makes the point free.
• 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.
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.

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 which try to insure 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.

## 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 maginification, 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 drag.

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.

### Multiple Views

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 view.

### Image Exporting

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.

## Constructions

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.
1. Create a new Construction by going to "File/New Construction".
2. Construct three points A, B, and C. These points will represent the ones given to the Construction.
3. 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.
4. Construct line segments AB and BC and their midpoints, D, and E.
5. 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.
6. Now construct the circle with center F that goes through the point A. This is the desired circumcircle.
7. 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.
8. 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!

### Non-Recursive Constructions

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 Construction List.

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 constructed.

• You will not be allowed to mark as Given objects which 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 which 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.

### Recursive Constructions

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 fractal.

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 which 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--which 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 which 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 which makes it invalid).