Element classes and construction methods for the Geometry Applet

Element classes and construction methods

for The Geometry Applet
version 2.2

There are eight different element classes: point, line, circle, polygon, sector, plane, sphere, and polyhedron. Only the first five classes are used in plane geometry. All can be used in solid geometry.

Each class has several construction methods. The eight tables below briefly explain these construction methods.

Special notes

1. Any time two points are needed in a construction, one line may be given instead. For instance, if AB has already been defined as a line that connects points A and B, then the triangle ABC can be defined by either of these:

	param="ABC;polygon;triangle;A,B,C"
	param="ABC;polygon;triangle;AB,C"

2. Those constructions that can only be used in solid geometry are indicated with a dark blue background color.

3. Optional data elements are indicated in the tables by square brackets. In particular, [z] indicates z is an optional integer, and [plane A] indicates A is an optional plane. The optional data elements are only used in solid geometry; they should always be omitted in plane geometry.

3a. When an optional third coordinate [z] is not specified, it is taken to be 0 so that the point lies in the xy-plane, that is, the plane of the screen.

3b. When an optional plane such as [plane A] is not specified, it is assumed to be the xy-plane.

4. Data elements come in three kinds: (1) integers, (2) points, and (3) other elements. These are indicated in separate lines in the tables. As long as the elements of each kind come in the right order, they will be interpreted properly. Elements of different kinds may be listed as you like.

Index of tables

Table 1: Element class point: Constructions: free, midpoint, intersection, first, last, center, lineSlider, circleSlider, circumcenter, vertex, foot, cutoff, extend, parallelogram, similar, perpendicular, proportion, invert, meanProportional, planeSlider, sphereSlider, angleBisector, angleDivider, fixed, lineSegmentSlider
Table 2: Element class line: Constructions: connect, angleBisector, angleDivider, foot, chord, bichord, perpendicular, cutoff, extend, parallel, similar, proportion, meanProportional
Table 3: Element class circle: Constructions: radius, circumcircle, invert, intersection
Table 4: Element class polygon: Constructions: square, triangle, quadrilateral, pentagon, hexagon, equilateralTriangle, parallelogram, regularPolygon, starPolygon, similar, application, octagon, face
Table 5: Element class sector: Constructions: sector, arc
Table 6: Element class plane: Constructions: 3points, perpendicular, parallel, ambient
Table 7: Element class sphere: Construction: radius
Table 8: Element class polyhedron: Constructions: tetrahedron, parallelepiped, prism, pyramid

Element class point

Construction method	Construction data	Description
free	integers x, y	a freely dragable point in the screen plane with initial coordinates (x,y,0)
midpoint	points A, B	the midpoint of a line AB
intersection	points A, B, C, D [plane E]	the intersection of two lines AB and CD in the plane E
intersection	points B, C plane A	the intersection of the plane A and the line BC
first	points A, B	the first end A of the line AB
last	points A, B	the last end B of the line AB
center	circle A	the center of the circle A
center	sphere A	the center of the sphere A
lineSlider	points A, B integers x, y,[z]	a point that slides along a line AB with initial coordinates (x,y,z)
circleSlider	circle A integers x, y,[z]	a point that slides along a circle A with given initial coordinates (x,y,z)
circumcenter	points A, B, C [plane D]	the center of a circle ABC passing through 3 points A, B, and C in the plane D
vertex	polygon A integer i	a vertex A_i of the polygon A₁A₂...A_n with index i
foot	points A, B, C	the foot of a perpendicular drawn from A to a line BC
foot	point A plane B	the foot of a perpendicular drawn from A to a plane B
cutoff	points A, B, C, D	the point E on a line AB so that AE = CD
extend	points A, B, C, D	the point E on a line AB so that BE = CD
parallelogram	points A, B, C	the 4th vertex D of a parallelogram ABCD given 3 vertices A, B, and C
similar	points A, B, D, E, F [planes C, G]	the point H so that triangle ABH in plane C is similar to triangle DEF in plane G
perpendicular	points A, B, [plane C]	the point D so that AD is equal and perpendicular to AB in plane C
	points A, B, D, E [plane C]	the point F so that AF is perpendicular to AB in plane C and equals DE
	points A, C, D plane B	the point E on the line perpendicular to plane B passing through A so that the distance from E to B equals CD
proportion	8 points A, B, C, D, E, F, G, H	the point I on GH so that AB:CD = EF:GI
invert	point A circle B	the image of a point A inverted in the circle B
meanProportional	6 points A, B, C, D, E, F	the point G on EF so that AB:CD = CD:EG
planeSlider	plane A integers x, y, z	a point that slides on the plane A with initial coordinates (x,y,z)
sphereSlider	sphere A integers x, y, z	a point that slides on the sphere A with initial coordinates (x,y,z)
angleBisector	points A, B, C [plane D]	The point at the intersection of the angle bisector of angle BAC and the line BC in plane D
angleDivider	points A, B, C [plane D] integer n	The point E on the line BC so that angle BAE is the n^th part of the angle BAC in plane D
fixed	integers x, y,[z]	the fixed point with coordinates (x, y, z)
lineSegmentSlider	points A, B integers x, y,[z]	a point that slides along within the line segment AB with initial coordinates (x,y,z)
harmonic	points B, C, D	the harmonic conjugate of B with respect to C and D

Element class line

Construction method	Construction data	Description
connect	points A, B	the line AB connecting two points A and B
angleBisector	points A, B, C [plane D]	the line AE bisecting angle BAC with E on BC in plane D
angleDivider	points A, B, C [plane D] integer n	the line AE with E on BC so that BAE is the n^th part of the angle BAC in plane D
foot	3 points A, B, C	the line AD drawn perpendicular to BC in the screen plane
foot	point A plane B	the line AD drawn perpendicular to plane B with the point D lying on B
chord	points A, B circle C	the intersection of the line AB in the circle C
bichord	circles A, B	the common chord connecting the two intersection points of the circles A and B
perpendicular	points A, B [plane C]	the line AD equal and perpendicular to AB in plane C
	points A, B, D, E [plane C]	the line AF perpendicular to AB in plane C equal to DE
	point A, C, D plane B	the line EF perpendicular to plane B passing through A equal to CD with E lying on B
cutoff	points A, B, C, D	the line AE equal to CD along the line AB
extend	points A, B, C, D	the line BE equal to CD so that A, B, and C are collinear with B between A and C
parallel	points A, B, C	the line AD parallel and equal to BC
similar	points A, B, D, E, F [planes C, G]	the line AH so that triangle ABH in plane C is similar to triangle DEF in plane G
proportion	8 points A, B, C, D, E, F, G, H	the line GI along GH so that AB:CD = EF:GI
meanProportional	6 points A, B, C, D, E, F	the line EG along EF so that AB:CD = CD:EG

Element class circle

Construction method	Construction data	Description
radius	points A, B [plane C]	the circle with center A and radius AB in the plane C
radius	points A, B, C [plane D]	the circle with center A and radius BC in the plane D
circumcircle	points A, B, C [plane D]	the circle passing through 3 points A, B, and C in the plane D
invert	circles A, B	the image of circle A inverted in circle B
intersection	spheres A, B	the intersection of spheres A and B

Element class polygon

Construction method	Construction data	Description
square	points A, B [plane C]	the square on a side AB in plane C
triangle	points A, B, C	the triangle ABC given 3 vertices A, B, and C
quadrilateral	points A, B, C, D	the quadrilateral ABCD given 4 vertices A, B, C, and D
pentagon	points A, B, C, D, E	the pentagon given 5 vertices
hexagon	points A, B, C, D, E, F	the hexagon given 6 vertices
equilateralTriangle	points A, B [plane C]	the equilateral triangle on a side AB in plane C
parallelogram	points A, B, C	the parallelogram ABCD given A, B, and C
regularPolygon	points A, B integer n	the regular polygon on a side AB given the number of vertices n
starPolygon	points A, B integers n, d	the star polygon on a side AB given the number of vertices n and the density d
similar	points A, B, D, E, F [planes C, G]	the triangle ABH in plane C is similar to triangle DEF in plane G
application	polygon A points B, C, D	the parallelogram equal to the given polygon A with one side BC and one angle BCD
octagon	8 points A, B, C, D, E, F, G, H	the octagon given 8 vertices
face	polyhedron A integer n	the n^th face of polyhedron A

Element class sector

Construction method	Construction data	Description
sector	points A, B, C [plane D]	the sector of a circle in plane D given the center A and two points B and C on the circumference
arc	points A, B, C [plane D]	the sector of a circle in plane D whose arc passes through the three points A, B and C

Element class plane

Construction method	Construction data	Description
3points	points A, B, C	the plane passing through points A, B, and C
perpendicular	points A, B	the plane passing through point A and perpendicular to line AB
parallel	plane A point B	the plane passing through point A and parallel to plane B
ambient	point A	the ambient plane of point A
ambient	circle A	the ambient plane of circle A

Element class sphere

Construction method	Construction data	Description
radius	points A, B	the sphere with center A and radius AB
radius	points A, B, C	the sphere with center A and radius BC

Element class polyhedron

Construction method	Construction data	Description
tetrahedron	points A, B, C, D	the tetrahedron given four vertices
parallelepiped	points A, B, C, D	the parallelepiped with three edges AB, AC, and AD
prism	polygon A points B, C	the prism with base A and side edges parallel and equal to BC
pyramid	polygon A point B	the pyramid with base A and apex B

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

Email: djoyce@clarku.edu
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