The coordinates of particle or a point is different for different frame of reference. A coordinate transformation is a transformation which established the relation between two coordinates of two different frame of reference. In this article’ Coordinate transformation and its Types ‘ we are going to discuss the basic idea of coordinate transformation. and also about the types of coordinate transformation.
Coordinate transformation and its Types
Coordinate transformation are of two types-
- Time-independent coordinate transformation
- Time-dependent coordinate transformation
Transformation equation for frame of reference having translation
Here, S is the stationary frame of reference. And S’ is moving frame of reference. From vector addition we have,
Then, r’=r−r₀ (1)
differentiating with respect to time t,
dr’/dt = dr/dt – dr₀/dt
v’ = v-0
(note:- because r₀ is constant here)
v’= v (2)
And, v= The velocity of the object with respect to S frame of reference.
Also, v’ = the velocity of the particle with respect to S’ frame of reference.
Again differentiating the equation (2) with respect to time t,
dv’/dt = dv/dt
That means acceleration remains constant.
Transformation equations in the frame of reference having uniform relative motion of translation
At time t=0
After time t=0 or at time t,
Let us suppose that S’ frame is moving with constant velocity with respect to S frame of reference .thus at time t,
We have ,r₀=vt (3)
(that is the distance travelled in time t , by moving the frame of reference )
From the diagram we have,
Now differentiate this equation with respect to time t,
And, here v’=velocity of the particle with respect to S’ frame of reference
V= the velocity of the particle with respect to S frame of reference
v= the velocity of S’ frame
Now, again differentiate with respect to time t,
That is a’=a
This shows that acceleration remains constant.
Transformation in an inclined frame of reference: for 2-D
S’ frame is inclined at angel θ, with S frame of reference. z and z’ axis are coincide with each other and their origins are same.
Now, take a point P, coordinates are (x,y) with respect to S frame of reference . And (x’,y’) with respect to S’ frame of reference.
Now, draw a perpendicular line from A to PD.
From diagram we have In ∆OAE –
∠OAE +∠EOA = 90⁰ (1)
∠OAE + ∠EAP =90⁰ (2)
After composition we get,
This shows that
∠EOA = ∠EAP = θ (3)
Cosθ = OE/OA
OE = x cosθ (4)
Similarly , sinθ = EA/OA
Sinθ = EA/x
x sinθ = AE (5)
Similarly from ∆FAP,
Cos θ = FA/AP
FA = y Cos θ (6)
Sinθ = PF/AP
Sinθ = PF/y
PF = y Sinθ (7)
From diagram we know,
x’ = PD = PF+ FD
x’ = y Sinθ+FD
Where, FD = OE
x’ = y sinθ+x cosθ (8)
Similarly from diagram,
y’ = PC = AF-AE
y’= y cosθ -x sinθ (9)
Equation (8),(9)and (10) are called transformation equations.
x’ = x cosθ + y sinθ
x’= x cos (x’ox) + y cos(90- x’ox)
x’ = x cos (x’ox) + y cos (x’oy) (11)
Similarly from equation (9)
(Note:- we know that -sinθ = cos (90 + θ)
θ = y’oy
= Cos (90 + y’oy)
= Cos (y’ox)
y’ = y Cos (y’ox) – x Sin (90 + x’ox)
y’ = y Cos (y’ox) + x cos (y’ox) (12)
Transformation in an inclined frame of reference : for 3-D
For 3D we can write
x’ = x Cos (x’ox) + y Cos( x’oy) + z Cos (x’oz) (1)
y’ = x Cos (y’ox) + y Cos (y’oy) + z Cos (y’oz) (2)
z’ = x Cos (z’ox) + y Cos(z’ oy) + z Cos(z’oz) (3)
We know Cos(x’ox)= iˆ.iˆ = a₁₁
Cos (x’oy) = iˆ.jˆ = a₁₂
Cos (x’oz) = iˆ.kˆ =a₁₃
Then equation will be,
x’ = x a₁₁ + y a₁₂+z a₁₃ (4)
y’ = x a₂₁ + y a₂₂+z a₂₃ (5)
z’= x a₃₁+ y a₃₂+z a₃₃ (6)
These are equation for transformation in 3-D
Now, differentiate these equations with respect to time t,
V’x = Vx a₁₁ +Vy a₁₂ + Vz a₁₃ (7)
V’y = Vx a₂₁ +V y a₂₂+Vz a₂₃ (8)
V’z= Vx a₃₁+V y a₃₂+Vz a₃₃ (9)
These are called transformation equations of velocity.
Similarly, for acceleration we Can write.
a’x = ax a₁₁+ aya₁₂ + aza₁₃ (10)
a’y = axa₂₁ + aya₂₂+aza₂₃ (11)
a’z = axa₃₁+ aya₃₂+aza₃₃ (12)
If we suppose that S is a inertial frame of reference. And there is no force applied on it then we have acceleration ax,ay,az =0
Thus from this equation (10),(11) and (12)
a’x,a’y and a’z =0
Hence we can say that S’ is also Inertial frame of reference.