Parametric and general equation of a geometrical object O
t
INPUT
(x, y, z)
INPUT
EQUATION
EQUATION
( x (t ) , y (t ) , z (t )) ∈ O
OUTPUT
OUTPUT
( x, y, z ) ∈ O

( x, y, z ) ∉ O
System of co-ordinates in E2
points
y
B = [b1 , b2 ]
b2
vectors
u
a1
a2
b1
A = [ a1 , a2 ]
u = AB = ( b1 − a1 , b2 − a2 )
B = A+ u
x
System of co-ordinates in E3
z
B = [b1 , b2 , b3 ]
points
vectors
u
A = [ a1 , a2 , a3 ]
y
x
u = AB = ( b1 − a1 , b2 − a2 , b3 − a3 )
B = A+ u
Parametric equations of a straight line
X
A
u
X = A+t ⋅u
in E3
in E2
A = [ a1 , a2 , a3 ]
A = [ a1 , a2 ]
u = ( u1 , u2 , u3 )
u = ( u1 , u2 )
x =
a1 + tu1
y = a2 + tu2
x =
a1 + tu1
y = a2 + tu2
z
= a3 + tu3
Parametric equations of a plane
X
u
A
v
A = [ a1 , a2 , a3 ]
u = ( u1 , u2 , u3 )
v = ( v1 , v2 , v3 )
X = A + su + t v
x =
a1 + su1 + tv1
y = a2 + su2 + tv2
z
=
a3 + su3 + tv3
General equation of a straight line in E2
ax + by + c = 0
If a straight line is given by two points A = [ a1 , a2 ] , B = [b1 , b2 ]
then its general equation can be expressed as
x − a1 y − a2
=0
b1 − a1 b2 − a2
General equation of a plane in E3
ax + by + cz + d = 0
If a plane is given by three points
A = [ a1 , a2 , a3 ] , B = [b1 , b2 , b3 ] , C = [ c1 , c2 , c3 ]
then its general equation can be expressed as
x − a1
y − a2
b1 − a1 b2 − a2
c1 − a1 c2 − a2
z − a3
b3 − a3 = 0
c3 − a3
General equations of a straight line in E3
A straight line n E3 is defined by the general equations of two
planes that intersect in that straight line.
a1 x + b1 y + c1 z + d1 = 0
a2 x + b2 y + c2 z + d 2 = 0
Given a general equation of a straight line ax + by + c = 0
u
v
u = k ( −b, a )
how can we determine a vector
u parallel to and a vector v
perpendicular to that straight
line ?
v = k ( a, b )
Given a general equation of a plane
ax + by + cz + d = 0
u
u = k ( a , b, c )
how can we determine a vector
u perpendicular to that plane ?
a1 x + b1 y + c1 z + d1 = 0
u = ( u1 , u2 , u3 )
a2 x + b2 y + c2 z + d 2 = 0
i
j
a1
a2
b1
b2
k
c1 = u1i + u2 j + u3 k
c2
Finding the coordinates of a
vector u parallel to the
straight line given by two
general equations of planes
Angle between two straight lines
v
u
u⋅v
cos ϕ =
u v
α
Angle between a plane and a straight line
u = ( u1 , u2 , u3 )
ϕ
ax + by + cz + d = 0
sin ϕ =
u1a + u2b + u3c
u12 + u22 + u32 a 2 + b 2 + c 2
Angle between two planes
ϕ
a1 x + b1 y + c1 z + d1 = 0
a2 x + b2 y + c2 z + d 2 = 0
cos ϕ =
a1a2 + b1b2 + c1c2
a12 + b12 + c12 a22 + b22 + c22
The distance of a point from a straight line or a plane
P = [ p1 , p2 , p3 ]
ax + by + c = 0
P = [ p1 , p2 ]
ax + by + cz + d = 0
d
d
d=
ap1 + bp2 + c
a +b
2
2
d=
ap1 + bp2 + cp3 + d
a 2 + b2 + c2
Special type equation of a straight line (1)
y
positive
ϕ
q
x
y = k⋅x+q
k = tan ϕ ,ϕ ≠
negative
π
2
Special type equation of a straight line (2)
y
x y
+ =1
a b
b
a
x
Special type equation of a plane
z
C
π ≈ ∆ABC
c
π
b
a
A
x
O
B
π≡
x y z
+ + =1
a b c
y
Canonical equations of a straight line in E3
u = ( u1 , u2 , u3 )
P = [ p1 , p2 , p3 ]
x − p1 y − p2 z − p3
=
=
u1
u2
u3
Bundle of straight lines
l1
l
l1 ≡ a1 x + b1 y + c1 = 0
l2 ≡ a2 x + b2 y + c2 = 0
l2
l ≡ λ1 ( a1 x + b1 y + c1 + d1 ) + λ2 ( a2 x + b2 y + c2 + d 2 ) = 0
Bundle of planes
π
π2
π1
π 1 ≡ a1 x + b1 y + c1 z + d1 = 0
π 2 ≡ a2 x + b2 y + c2 z + d 2 = 0
π ≡ λ1 ( a1 x + b1 y + c1 z + d1 ) + λ2 ( a2 x + b2 y + c2 z + d 2 ) = 0
Bunch of planes
C
π 1 ≈ ∆PAB, π 2 ≈ ∆PBC ,π 3 ≈ ∆PCA
A
B
P
π 1 ≡ a1 x + b1 y + c1 z + d1 = 0
π 2 ≡ a2 x + b2 y + c2 z + d 2 = 0
π 3 ≡ a3 x + b3 y + c3 z + d3 = 0
π ≡ λ1 (π 1 ) + λ2 (π 2 ) + λ3 (π 3 ) = 0