Plane Cross
Sections of a Cube
Ilda Reis
Representing Plane Cross Sections of a Cube
in a Bidimensional Coordinate System
An application in GeoGebra 2D
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
International GeoGebra Institute Conference
Warsaw 21 - 23 September, 2012
Joint work with Edite Cordeiro
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Outline
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Introduction
Implementation of a 3D dynamic axial system in GeoGebra
Cross section of a cube
Relationship between cross section shapes and position of
the cutting plane
Conclusions
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Introduction
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
A cross section of a cube is the polygon obtained by the
intersection of that cube and a given plane.
According to Portuguese mathematics high school curriculum it is
expected that students be able to represent geometric solids on a
plane and to draw cross sections of a cube.
The main goals of this work are:
I
To represent 3D objects in a 2D coordinate system;
I
To draw the plane cross section of a cube given a plane
I
I
3 points on the cube edges
3 points on the cube faces
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Implementation of a 3D dynamic axial system in
GeoGebra
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
A new 3D axial system is build and projected onto a plane to
simulate the representation of space objects. This new system is
controlled by parameters that enables the visualization and
manipulation of these objects in different perspectives.
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
3D rotation matrices
Plane Cross
Sections of a Cube
Ilda Reis
Let O = (0, 0, 0) be the origin and {i, j, k} = {(1, 0, 0), (0, 1, 0),
(0, 0, 1)} be 3 vectors along the coordinate axes of Oxyz.
Rotating Oxyz to γ degrees about the z-axis is equivalent to
perform a rotation on the plane Oxy and to maintain the
z-coordinate. This linear map is represented by the following
matrix:


cos(γ) − sin(γ) 0
Rz (γ) =  sin(γ) cos(γ) 0  .
0
0
1
Similarly, let Rx (α) and Ry (β) be the 3D rotation matrices of α
degrees about the x-axis, and β degrees about the y -axis of the
Oxyz, respectively.
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
An object, when subjected to a sequence of these 3
rotations, acquires a specific orientation, that is rotation
sequence dependent.
References
Projective Geometry and 3D Visualization
Plane Cross
Sections of a Cube
Ilda Reis
Let p : R3 → R2 be the orthogonal projection of the system Oxyz
onto the plane Oyz, defined by the matrix
0 1 0
Mp =
.
0 0 1
The linear transformation defined by
M = λMp Rx (α)Ry (β)Rz (γ),
λ>0
maps {i, j, k} to {u, v , w } = {Mi, Mj, Mk} from the plane Oyz.
Therefore, a point P = (x, y , z) ∈ R3 is represented on the
projected coordinate system {O, u, v , w } by the linear combination
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
P = xu + yv + zw .
From now on, the axial system to be used is {OT , u, v , w }, where
OT is an arbitrary point in R2
References
Creating a projected axial coordinate system
Plane Cross
Sections of a Cube
Ilda Reis
The implementation process to simulate a 3D object is
time-consuming.
Geogebra allows user defined tools to reduce the time required for
implementing a procedure.
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Cross section of a cube
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
The polygon obtained is plane orientation-dependent.
A cross section is drawn using properties from elementary
Euclidean geometry, namely
I
If a plane contains two points of a line, then that plane
contains the whole line;
I
If two distinct planes intersect, then their intersection is a
line;
I
If a plane intersects two parallel planes, then the intersection
is two parallel lines.
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Notation
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Let
I
V = {V1 , V2 , . . . , V8 } be the cube’s vertices set;
I
A = {a1 , a2 , . . . , a12 } be the cube’s edges set
a5 = [V5 , V6 ], a6 = [V6 , V7 ], a7 = [V7 , V8 ], a8 = [V5 , V8 ],
Relationship
between cross
section shapes and
position of the
cutting plane
a9 = [V1 , V5 ], a10 = [V2 , V6 ], a11 = [V3 , V7 ] and a12 = [V4 , V8 ]).
Conclusions
F = {F1 , F2 , . . . , F6 } be the cube’s faces set
References
(a1 = [V1 , V2 ], a2 = [V2 , V3 ], a3 = [V3 , V4 ], a4 = [V1 , V4 ],
I
Given 3 points on its
edges
Given 3 points on its
faces
(F1 = ConvexHull(V1 , V2 , V3 , V4 ), F2 = ConvexHull(V5 , V6 , V7 , V8 ),
F3 = ConvexHull(V1 , V2 , V5 , V6 ), F4 = ConvexHull(V2 , V3 , V6 , V7 ),
F5 = ConvexHull(V3 , V4 , V7 , V8 ) and F6 = ConvexHull(V1 , V4 , V5 , V8 ))
Cross section: A ∈ a1 , B ∈ a2 and C ∈ a11
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Cross section for 3 given points on edges
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Cross section: A ∈ F1 , B ∈ F2 and C ∈ F3
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
Let B1 and C1 be the projection of B and C onto F1 ,respectively.
D1 = BC ∩ B1 C1 belongs to the plane that contains F1
AD1 is on the plane defined by ABC
R and S are two of the three desired points.
References
Cross section for 3 given points on faces
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Relationship between cross section shapes and
position of the cutting plane
Four geometric shapes are possible: triangles, quadrilaterals,
pentagons and hexagons.
A triangle cross section is obtained when the cutting plane cuts
three edges meeting at a vertex.
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Type of triangles
Cutting plane position
Equilateral
Isosceles
Scalene
parallel to two face diagonals
parallel to one face diagonals
not parallel to any face diagonals
All of them are acute triangles.
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Relationship between cross section shapes and
position of the cutting plane
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
A quadrilateral cross section is obtained when the cutting plane
cuts four faces.
It is possible to obtain squares, rectangles, parallelograms,
trapeziums and rhombus.
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
A pentagon is obtained when the plane cuts five faces of the cube.
This polygon can not be regular.
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
An hexagon is obtained when the cutting plane cuts all faces of
the cube. This polygon is regular when the vertices are the
midpoints of the supporting edges.
References
Conclusions
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
The cross section of a cube was drawn given three points on its
edges.
Given three points on its faces, it was able to find three points on
its edges capable to determine each cross section.
For this work it is worthy to highlight the importance of Linear
Algebra concepts for the representation of 3D objects on the plane
which allowed the visualization and manipulation of the built
objects in different perspectives.
This kind of mathematical dynamic resources can help students to
better understand some mathematical concepts.
Implementation of
a 3D dynamic axial
system in
GeoGebra
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
References
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Jeong-Eun Park, Young-Hyun Son, O-Won Kwon, Hee-Chan
Yang, Kyeong-Sik Choi, Constructing 3D graph of function
with GeoGebra(2D), First Eurasia Meeting of GeoGebra 1
(GeoGebra Institute of Ankara) 46-55, 2010.
S. Lang, Intruduction to Linear Algebra, 2 Ed., Springer, 2008.
Judith e Markus Hohenwarter, Introduction to GeoGebra4 .
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References
Plane Cross
Sections of a Cube
Ilda Reis
Introduction
Implementation of
a 3D dynamic axial
system in
GeoGebra
Thank you for your attention.
Representing R3 on
the plane
Creating a projected
axial coordinate
system onto the plane
Cross section of a
cube
Given 3 points on its
edges
Given 3 points on its
faces
Relationship
between cross
section shapes and
position of the
cutting plane
Conclusions
References