A POint, Line, and Plane on Space

A Point, Line, and Plane on Space
Analysis and Model
There are three original element in goemetry, they are point, line, and plane. The third element are undefined element.
A point can be imagined look like a position on space. A point hasn’t length and thick. Trace of needle puncture or track of pencil tip can be assumption as a model of a point. A point can be representatived by a dot which given named by a capital letter.
A line can be assumption as a set of point in a infinite long row, and it haven’t width. A line can be representatived by a rope of thread be sprawled.
A plane can be assumption as a set of point in a infinite tightly row and it haven’t width.
C
D
E
B
A
α
F
Picture 1
On picture 1, α-plane contain A, B, C, and D points. We can say that the fourth points located on α-plane. and on α-plane. Based on picture.1 we can say that :
B ∈ α → means B-point on α-plane
C ∈ → means C-point on
∈ α → means on α-plane
= -plane, etc.
The position of point, line, and plane on space
The position of two points
Definition : two coincide points means two same points.
A
B
Picture.2
The position of point and line
Definition : colinear points are points on a line.
C
A
g
h
B
D
Picture.2
Based on picture.3, we can see that A and B are two colinear points, because both of them on h-line. Reciprocally, C and D are two colinear points, because they are on g-line. But A and C are not colinear points because they are not in a line.
The position of point and plane
A point can be located on a plane or not. If A-point on α-plane, we can say that α-plane through A-point or A-point on α-plane.
Axiom : any three points on at least one plane
any three non-colinear points accurately on one plane
Definition : coplanar
Coplanar points if and only if there is a plane which hold them.
P
Q
α
R
S
Picture.3
Based on picture.4, we can see that P, Q, and R are non-colinear points, however they are on one plane, it is α-plane. So, we can say that P, Q, and R re the third coplanar points. While, S-points is not on α-plane. Therefore, P, Q, R, and S are non-coplanar points.
The Position of two lines
Two lines can be assumption on a plane or not. If two lines are on a plane, they may intersect at one point or parallel. If two lines are not on a plane, we can say that they are cross each other.
Definition : ( Parallel and cross-lines)
Two different lines are parallel if and only if they are coplanar and they don’t intersect at a point.
Two different lines are cross each other if and only if they are coplanar.
D
gefinition : if there ar two different lines intersect at a point, so they are precise on a plane.
k
m
h
Q
R
n
P
Picture.4
α
Based on picture.5 : h, k, and m-lines are coplanar because they are on a plane, it is α-plane. h and k are parallel lines and they are on a plane. k and m are intersect at Q-point and they are on a plane. Likewise, h and m intersect at P-point and they are on a plane. n is not on α-plane. g-line pierces α-plane precise at a point, it is R-point. We also can say that g-line is not on α-plane. but m-line is on α-plane. therefore g and m are cross-lines. G-line is not on α-plabe but k is on α-plane. Therefore g and k are cross-lines.
The position of line and plane
I
g
gf there are a line and a plane, there are some possibility, such as the line pierces the plane, the line is parallel with the plane, or the line is on the plane.
h
R
α
α
Picture.5
Picture.6
g
α
Picture.7
Based on picture.5, g-line pierces α-plane. g-line and α-plane are interset each other if they have precise a league point. g-line pierces α-plane precise at a point, it is R-point. so we can say that R is intersection point of g-line and α-plane.
Based on picture.6, g-line is not on α-plane and h-line is on α-plane. g and h are parallel. So we can say that g-line is parallel with α-plane. g-line and α-plane are parallel if they are not coalesce at any point.
Based on picture.7, whole of g-line is on α-plane. means all points of g-line are on α-plane.
The position of two planes
If there are two planes, so there are two possibility, such as they will intersect or they are parallel each other.
β
(α,β)
α
α
β
Picture.8
Picture.9
Picture.8 : α and β are intersect each other if they are coalesce precise at a line. The coalition line is named by intersection line of α and β-planes. So, we can say that (α,β)-line is a set of all points which on α-plane and β-plane.
Picture.9 : α and β are parallel, if they don’t coalesce at any point.
T
she position of three planes
s
P
α
α
γ
β
β
γ
s
Picture.10
Picture.11
α
γ
β
Picture.12
α
β
γ
α
β
γ
Picture.13
Picture.14
Picture.10 : γ-plane intersect s-line precises at a point, it is P-point.
Picture.11 : γ-plane contain all parts of s-line. It means that α, β, and γ-planes are intersect at a line, it is s-line.
Picture.12 : γ-plane is parallel with s-line. s-line is intersection point of α-plane and β-plane, and s-line is parallel with γ-plane, so (α,γ)-line and (β,γ)-plane are parallel with s-line. so, s-line ∥ (α,β)-line ∥ (β,γ)-line.
Picture.13 : γ-plane is intersect α-plane. α∥β, so γ-plane is intersect β-plane. so, (α,γ)-line ∥ (β,γ)-line.
Picture.14 : γ ∥ α, because α ∥ β and γ ∥ α, so γ ∥ β. We can concluse that α-plane ∥ β-plane ∥ γ-plane.
E
G
Hxercises.
E
F
C
D
A
B
Picture.15
Answer these questions based on information of picture.15 !
1). Coplanar lines such as :
a.
b.
c.
d.
answer : c
2). The plane which parallel with BCGF-plane is :
a. ABFE-plane
b. ADHE-plane
c. ABCD-plane
d. EFGH-plane
answer : b
The discussion has been hold in Diyani Arif Setyorini’s rent house at Gejayan Street after we got the lectures, on Thursday, 18 December 20008 about 4 p.m.. I’ve used a literature of first semester, its title is “Suplemen Bahan Kuliah Ilmu Ukur Analitik Ruang“. Bluffy, she took my papers, then read it. She could conceived all parts of the material. Without I explained to her, she could answered two questions correctly. According to me, she has good imagination to ‘make sketches’ of each condition. So, she hasn’t got difficulties.