Translation Transformation Calculator
A translation is a rigid transformation that “slides” every point in the plane
by the same vector. It does not rotate, stretch, or reflect the figure.
This means translations preserve distances, angles, and areas.
1) Translation of a point
Let the translation vector be
\[
\mathbf{v}=(d_x,d_y).
\]
If \(P=(x,y)\), then the translated point \(P'\) is:
\[
P'=(x',y')=(x+d_x,\;y+d_y).
\]
2) Translation of a shape (vertices)
A shape (segment, triangle, polygon, etc.) is typically defined by vertices
\(P_1,P_2,\dots,P_n\). A translation applies to each vertex:
\[
P_i=(x_i,y_i)\quad\longrightarrow\quad P_i'=(x_i+d_x,\;y_i+d_y).
\]
After computing all translated vertices, connect them in the same order to get the shifted shape.
3) Matrix form (homogeneous coordinates)
Translations are conveniently expressed using homogeneous coordinates. Write a point as
\(\begin{bmatrix}x\\y\\1\end{bmatrix}\). Then:
\[
\begin{bmatrix}x'\\y'\\1\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & d_x \\
0 & 1 & d_y \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}x\\y\\1\end{bmatrix}.
\]
This is useful in graphics pipelines where multiple transformations (rotate, reflect, translate) are combined by multiplying matrices.
4) What does a translation preserve?
-
Distance: for any two points \(A,B\),
\[
|AB| = |A'B'|.
\]
-
Angles: if two segments meet at an angle \(\theta\), the translated segments meet at the same \(\theta\).
-
Area and perimeter: polygons keep the same area and perimeter after translation.
-
Parallelism and collinearity: lines remain parallel, and points remain on the same line.
5) Vector magnitude and direction (extra insight)
The “amount” of shift is the vector length:
\[
\|\mathbf{v}\|=\sqrt{d_x^2+d_y^2}.
\]
The direction (angle from the positive \(x\)-axis) is:
\[
\theta=\operatorname{atan2}(d_y,d_x).
\]
(The calculator reports this angle in degrees.)
6) Worked example
Translate \(P=(2,3)\) by \(\mathbf{v}=(4,-1)\).
Apply the rule \(x'=x+d_x,\;y'=y+d_y\):
\[
x' = 2+4 = 6,\qquad y' = 3+(-1) = 2.
\]
So the new point is:
\[
P'=(6,2).
\]
7) Common input tips
-
For polygons, enter one vertex per line, such as
(0,0) or 0,0.
-
Use
* for multiplication (e.g., 2*pi).
-
A translation does not depend on where the point is—it always adds the same \((d_x,d_y)\).
In the diagram, the “before” figure and the “after” figure are drawn with the same shape.
If slide animation is enabled, the calculator shows an intermediate position
\(P(t)=P+t\mathbf{v}\) for \(0\le t\le 1\) to visualize the motion.
