Appendix A Dr.Geo Glossary

this Glossary assumes that the learner understands the vocabulary and constructions of elementary Euclidean geometry, and aims to provide simple and clear explanations of terms used in geometry beyond that level, algebra, Dr.Geo, and Smalltalk sufficient for the beginner to proceed with, and also in some significant cases very correct explanations that will help the more advanced learner to understand Dr.Geo and Smalltalk further. Full understanding requires other tools, such as a Smalltalk reference manual and study of Dr.Geo source code, or even extensive study of Computer Science.

Complete understanding is not given to mere mortals, and there are theorems implying that it is actually a self-contradictory idea. Besides, where would the fun be in having nothing left to learn?

abcissa In coordinate geometry, the x coordinate of a point. In Dr.Geo, it can also be the difference between the x coordinates of the initial and final points of a vector. See also curvilinear abscissa.

API Application Programming Interface; in Dr.Geo, methods usable in scripts to apply to math items, from the class hierarchy DrGMathItem.

argument A value provided to a function, assigned to a variable that it can act on. In Smalltalk, however, messages are sent to objects rather than applying functions to values. One can view the recipient of a message as an argument of the method invoked by the message. Sometimes called a parameter of a function.

array An indexable sequence of Smalltalk objects created with either of two notations. For example, {} and #() are the empty array; {1} and #(1) are an array with one element, the integer 1; {’a’ . 2} and #(’a’ 2) are an array containing two elements; and so on.

background process Define

balloon help Explanations of objects in a user interface that appear when the mouse briefly points to the object.

block In Smalltalk syntax, a segment of Smalltalk code, optionally including a declaration of local variables, enclosed in brackets.

block closure A Smalltalk object of class BlockClosure representing a program, defined by a block, that is not an object method, together with the environment it executes in, so that the closure can be a parameter of a control structure such as whileTrue:, which is a method of the BlockClosure class, or ifTrue: which is a method of the Boolean class.

browser See class browser.

change set In Smalltalk, a change set is created by saving all new and modified class and method definitions. Although change sets can be accessed via the Extended search submenu in a workspace, they are not relevant to normal use of Dr.Geo.

class In Smalltalk, an object representing a type of object (class members) together with the behavior of its members (methods).

class browser A Smalltalk development tool that displays the classes and methods of Smalltalk objects in an organized manner, and provides access to detailed information on all objects and code. Also known as System browser.

class comment A text string in a class definition usually intended to provide information on the purpose of a class (class intention), any variables it exposes, and how to use the class and its methods (class collaborations, class API).

closure See block closure.

constraint The relationship between geometric elements in which a change in the position, orientation, or size of one affects the properties of another, or in which properties of some elements prevent some kinds of change to another. For example, after constructing the intersection of two lines, moving one of the lines generally moves the intersection, but the intersection cannot be moved directly.

constructionist learning The terminology of education is vexed and confusing, with the same word used in multiple meanings, and very similar words used in very different ways, all compounded by deep ignorance and misunderstandings about human nature.

Seymour Papert defined constructionism in a proposal to the National Science Foundation entitled “Constructionism: A New Opportunity for Elementary Science Education” as follows: "The word constructionism is a mnemonic for two aspects of the theory of science education underlying this project. From constructivist theories of psychology we take a view of learning as a reconstruction rather than as a transmission of knowledge. Then we extend the idea of manipulative materials to the idea that learning is most effective when part of an activity the learner experiences as constructing is a meaningful product." Geometric constructions and computer programs are but two of many such products.

control structure A method of an appropriate class for determining whether or how often to execute a block of code, such as ifTrue: or whileTrue:.

Creative Commons An organization devoted to sharing of content in much the same manner as Free/Open Source Software, providing licenses that guarantee that a work can be copied freely (CC), and if desired translated and improved (Sharealike), as long as the results are published under the same license. It provides other licensing options as well.

curvilinear abscissa In Dr.Geo, the parametric coordinate of a free point on a curve in the range [0 ; 1]. For circles, arcs, and polygons, this is proportional to arc length. For straight lines, and for loci based on straight lines, it is based on the arctangent of the distance from a base point, mapping [-infinity ; infinity] to [0 ; 1]. One of the points defining the line is taken to be its origin, with curvilinear abscissa 0.5, while a point on the line with curvilinear abscissa 0 or 1 is at infinity.

data pane The area at the left in a Dr.Geo sketch window, listing the items in the sketch in the order of their creation, and giving some essential properties of the items.

debugger A Smalltalk tool for investigating programs, which can be used to find and correct errors or to explore the working of Smalltalk code.

degree A measure of angles, with 360 degrees in a circle, 180 degrees in a straight angle, and 90 degrees in a right angle.

development environment A set of tools for creating, inspecting, and maintaining software.

domain The set of values on which a function is defined.

dynamically typed Of a programming language, allowing the type of a variable to be changed during program execution, or allowing a data structure to be expanded or shrunk. In Smalltalk, objects of differen classes can be assigned to the same variable at different times.

free point A point that the user can drag freely in the workspace or on a curve. Contrast with constrained point, defined by its relations to other sketch objects, such as the intersection of two lines.

Free Software Software provided under a license that protects the rights of the user, especially the four Software Freedoms defined at http://www.gnu.org/philosophy/free-sw.html:

1. 0 The freedom to run the program, for any purpose.
2. 1 The freedom to study how the program works, and change it so it does your computing as you wish. Access to the source code is a precondition for this.
3. 2 The freedom to redistribute copies so you can help your neighbor.
4. 3 The freedom to distribute copies of your modified versions to others. By doing this you can give the whole community a chance to benefit from your changes. Access to the source code is a precondition for this.

functional Of interactive software, requiring commands to be explicitly invoked at each use, as opposed to modal, where a command can be applied multiple times after being selected once. Dr.Geo has elements of both, and can be toggled between them with the single/multiple creation button.

global variable A variable accessible anywhere. Contrast with local variable and instance variable.

homothety Scale to larger or smaller size. For example, make similar triangles.

instance variable A variable declared in the header of a method as a local variable, accessible only within that method.

local variable A variable whose value can be accessed or changed only in some restricted context, such as the methods of a single class, or within a single method. Contrast with global variable.

lock Fix a Dr.Geo point in place in a sketch so that it cannot be moved by clicking and dragging. It can still be relocated under program control or by editing its coordinates in the property dialog.

hide Make a Dr. Geo object invisible and unselectable in the sketch. Hidden objects appear in light gray and can be selected when editing object styles.

infinity In geometry, a point at infinity is defined as the intersection of parallel lines. In algebra, an infinity can be defined as the limit of a divergent sequence, as 1/0, as tan(pi/2) and in other ways. Dr.Geo allows for points and even lines at infinity, but does not handle algebraic infinities well.

inspector In Smalltalk, a tool for viewing properties of an object.

label The name of an object, displayed alongside it. The user can create labels in the object style editor.

locus A curve or set of points defined by constraints such as distance from given points or lines. For example, the locus of a point equidistant from two rays is the bisector of the angle between the rays. Plural loci.

macro In many programming systems, macro processing means making text substitutions in source code. In Dr.Geo a macro is an encapsulated construction specifying a set of objects as its starting point, and another set of objects as results. The construction must uniquely determine the outputs from the given inputs. Where this is the case, Dr.Geo determines how to carry out the construction and records the procedure for reuse. Macros are built using tools accessed in the macro construction tab.

math item Object in a class from the hierarchy DrGMathItem, including points and curves of various kinds, numeric values, and others.

method In object-oriented programming, a program associated with a class specifying some behavior of the object type of that class (instance method) or of the class itself (class method). The name of the method is called its selector.

method finder A Smalltalk tool for searching for methods based on their names, their associated classes, their program text, examples of use, or pragmas.

method source The code defining a Smalltalk method.

method string Literal string within method code.

modal Of software, having distinct states or modes in which particular functions can be applied repeatedly, as opposed to stateless or functional, where the function has to be specified for each use. Dr.Geo has elements of both, and can be toggled between them with the single/multiple creation button.

name In the object style properties dialog for Dr. Geo objects, a label to be displayed with the object.

norm Length of a vector.

object In Smalltalk, a member of a class whose behavior is specified by methods defined in that class.

object-oriented In programming languages, using object definitions rather than data types, where objects are defined in classes by the methods that act on them, and classes can inherit methods from other classes, usually in a tree-structured hierarchy. The class structure of Pharo Smalltalk can be examined using the hierarchy function in the class browser in Dr.Geo.

Open Source Software published under the Debian Open Source guidelines or any other license that requires making source code available to all users without charge or other restrictions. Similar to Free Software, but there are differences in the terms of the various licenses.

ordinate In coordinate geometry, the y coordinate.

pane An area in a Smalltalk window reserved for a particular purpose.

parameter Another name for a function argument. See also parametric curve.

parametric curve A curve defined by functions of a variable other than the coordinates. For example, x = cos(t), y = sin(t), where x and y are the coordinates of a point, defines a circle with t as the parameter.

Pharo Smalltalk The version of Smalltalk used to implement Dr.Geo.

pin Click in a sketch window to place an object at the point selected.

plane In Dr.Geo, an extended Euclidean plane, where parallel lines are equidistant, and in principle intersect in the same point at infinity in both directions. Dr.Geo does not handle points and lines at infinity consistently, where a line at infinity joins two points at infinity. This is in contrast with the standard Euclidean plane, which does not have points at infinity; with non-Euclidean planes where there are no equidistant straight lines and there may be points at or beyond infinity; and with the projective plane, where all pairs of lines intersect, but there is no concept of distance or length. It would be possible to use homogeneous coordinates in a projective plane to make Dr.Geo handle infinities consistently, but explaining them is beyond the scope of this glossary. Note that it is possible to model and explore all of these geometries within Dr.Geo, and that an Omar (Our Most Assiduous Reader, in the lore of Conway game theory) could reconstruct Dr.Geo to support any or all of them (and more) directly.

pointBlockItem a DrGPointBlockItem, defined as a "Point item defined by a block closure". It is a subclass of DrGPointCoordinateItem, which is a subclass of DrGPointItem, so that it responds to the normal point methods.

pragma A system for tagging methods, used by the method finder but not relevant in Dr.Geo.

protocol The set of categories for the methods in a class, displayed in the class browser.

radian A measure of angles based on the lengths of arcs in a unit circle with circumference 2pi. 1 in radian measure is the angle subtending an arc of length 1 in this circle, or more generally the radius in any circle. Commonly used in symbolic form in mathematics, where a right angle is represented as pi/2. For purposes of arithmetic, angle measure in degrees is usually more convenient.

reference Within Smalltalk, any occurrence of a given class name in any class definition or method code. A search for "references to it" using Extended Search on the right-click menu in a workspace opens a Users tool. See Smalltalk documentation for details.

reflective In a programming language, providing tools for examining objects by methods other than reading source code files.

script A Smalltalk method created in the DrGeoUserScripts class that can be called using the Use a script button in the Numerics tab or the Use a script command in the Numerics menu. The result of a script can be pinned in a sketch, so that the script is saved and reloaded with the sketch.

selector In Smalltalk, the name of a method. It may contain more than one part, in which case each part will end with a colon. For example, distance: to:.

sender For a given selector, a class that has a method of that name.

sketch A geometric figure defined using Dr. Geo commands, including constraints, formatting, and other properties of points, lines, curves, and compound elements. Macro constructions, equations, and pinned scripts are also parts of a sketch.

sketch pane The area in a Dr.Geo sketch window below the toolbars where geometric, numeric, algebraic, and script items are placed to form the sketch.

Smalltalk An object-oriented, graphical programming language and environment designed by the Learning Resources Group at Xerox PARC (Palo Alto Research Center), the model for all modern Graphical User Interfaces (GUI) after the Apple Lisa and Macintosh were based on it.

Smalltalk-80 The first publicly-released version of Smalltalk.

stateless Of interactive software that is not modal, that is, where commands do not change the state of the application so that only certain commands are active until the user changes to another mode.

style Properties of a sketch object, such as name (also called label), colour, shape, size, and the states hide/display and lock/unlock.

system browser The class browser.

type In programming languages generally, the type of a variable defines the number of bits used to represent it plus any internal structure, and the functions that apply to it. In Smalltalk, object types are defined as classes, including the methods that apply to objects in each class, and do not imply any particular representation in memory or in storage.

value Of a variable, the object it refers to at the time, if any. Of a function, the object it returns when called with particular arguments, if any.

variable A name that functions like a pronoun in human languages, being able to refer to different objects in different classes at different times. Variables are implemented as objects, just like everything else in Smalltalk. Making a variable refer to a different object is called assigning that object to the variable as its value.

visibility The property of displaying in the sketch or being hidden.

workspace A Smalltalk window for entering, executing, and debugging Smalltalk code.