Image (Transformation)
Geometry & MeasurementThe image is the resulting figure after a transformation has been applied to the original figure (preimage).
Definition
The image is what a shape looks like after you apply a transformation. The original shape is the preimage, and the result after moving, flipping, or resizing is the image.
Example
If you reflect triangle $ABC$ across the y-axis, the original triangle $ABC$ is the preimage and the new triangle $A'B'C'$ is the image. The prime symbol ($A'$) shows a point is in the image.
Key Insight
The image is always labeled with prime marks ($A'$, $B'$, $C'$) to show it came from the preimage ($A$, $B$, $C$). Think of "prime" as "after transformation." Reading the prime mark tells you which original point it came from.
Definition
In the context of geometric transformations, the image of a figure $F$ under transformation $T$ is the set $T(F) = \{T(P) : P \in F\}$. Each point $P$ in the preimage maps to a corresponding image point $P' = T(P)$. For rigid motions, the image is congruent to the preimage; for dilations, it is similar.
Example
Under translation $T(x,y) = (x+2, y-3)$, the image of segment from $(1,4)$ to $(5,4)$ is the segment from $(3,1)$ to $(7,1)$. The image has the same length ($4$ units) but a new position.
Key Insight
The word "image" in transformation geometry is the same as in function notation. If $f(x) = 2x$, the image of $x=3$ is $f(3)=6$. Similarly, if $T$ is a geometric transformation, the image of point $A$ is $T(A)$. Geometry and algebra use identical language.
Definition
For a transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$, the image of set $S$ is $T(S) = \{T(x) : x \in S\}$. For linear maps, the image (range) of $T$ is the column space of the matrix. For isometries, the image is congruent to the preimage (same measure). For general differentiable maps, the image near a regular point is approximated by the image of the linearization (Jacobian).
Example
The image of the unit circle under the linear map $T(x,y) = (2x, y)$ (horizontal stretch by $2$) is the ellipse $\{(x,y) : (x/2)^2 + y^2 = 1\}$. The Jacobian of $T$ is the matrix $\begin{bmatrix}2&0\\0&1\end{bmatrix}$, with $\det = 2$, so areas scale by factor $2$.
Key Insight
The image of a smooth curve under a differentiable map is determined locally by the Jacobian. If the Jacobian is invertible (nonzero determinant) at a point, the inverse function theorem guarantees the image is locally a smooth curve, preserving the topology of the original near that point.