Origin
Geometry & MeasurementThe origin is the point (0, 0) where the x-axis and y-axis intersect on the coordinate plane.
Definition
The origin is the center point of the coordinate plane where the x-axis and y-axis cross each other. Its address is $(0, 0)$.
Example
If you draw a coordinate grid on paper, the origin is the point right in the middle where the two lines meet. All other points are measured by how far they are from the origin.
Key Insight
The origin is the "starting point" for all coordinate measurements. It is like the address $(0, 0)$ on a map: everything else is described by its distance and direction from this center point.
Definition
The origin is the point $(0, 0)$ in the Cartesian coordinate system, the unique point where the x-axis and y-axis intersect. It is the additive identity of the vector space $\mathbb{R}^2$: any vector added to the zero vector $(0, 0)$ is unchanged.
Example
A rotation is centered at the origin by default. Rotating point $(3, 0)$ by $90$ degrees counterclockwise about the origin gives $(0, 3)$. Scaling from the origin by factor $2$ maps $(3, 4)$ to $(6, 8)$.
Key Insight
The origin is the fixed point of all dilations centered there. When the center of a dilation is the origin, coordinates simply multiply by the scale factor: $(x, y)$ maps to $(kx, ky)$.
Definition
The origin is the zero vector $0$ in $\mathbb{R}^2$, the identity element of the additive group $(\mathbb{R}^2, +)$. In projective geometry, the origin loses special status as all points become equivalent under projective transformations. In complex analysis, the origin $z = 0$ is often a critical point or singularity of functions like $1/z$.
Example
A linear map $T: \mathbb{R}^2 \to \mathbb{R}^2$ always satisfies $T(0) = 0$: linear maps always fix the origin. Affine maps are linear maps plus a translation and do not necessarily fix the origin, making them more general.
Key Insight
The distinction between linear and affine transformations (whether the origin is fixed) is fundamental in graphics, robotics, and physics. Changing the origin (a translation) can simplify problems by exploiting symmetry.