Affine transformation

From WikiMD.com - Food, Medicine & Wellness Encyclopedia

Affine Transformation[edit | edit source]

An affine transformation is a function between affine spaces which preserves points, straight lines, and planes. Sets of parallel lines remain parallel after an affine transformation. An affine transformation can be constructed using a linear transformation followed by a translation.

Affine transformations are fundamental in computer graphics, image processing, and computer vision, as they can represent any combination of translation, scaling, rotation, and shearing.

Mathematical Definition[edit | edit source]

An affine transformation in two-dimensional space can be represented by the equation:

\( \mathbf{y} = \mathbf{A} \mathbf{x} + \mathbf{b} \)

where \( \mathbf{A} \) is a linear transformation matrix, \( \mathbf{x} \) is the input vector, \( \mathbf{b} \) is the translation vector, and \( \mathbf{y} \) is the output vector.

In matrix form, this can be written as:

\[

\begin{bmatrix} y_1 \\ y_2 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & b_1 \\ a_{21} & a_{22} & b_2 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ 1 \end{bmatrix} \]

Types of Affine Transformations[edit | edit source]

  • Translation: Shifts every point of a shape in the same direction by the same distance.
  • Scaling: Enlarges or diminishes objects; the scale factor is the same in all directions.
  • Rotation: Rotates objects around a fixed point.
  • Shearing: Slants the shape of an object.
  • Reflection: Flips objects over a line.

Properties[edit | edit source]

Affine transformations preserve:

  • Collinearity: Points that lie on a line continue to be collinear.
  • Ratios of distances: The midpoint of a line segment remains the midpoint after transformation.
  • Parallelism: Parallel lines remain parallel.

Applications[edit | edit source]

Affine transformations are widely used in:

Related Pages[edit | edit source]

Gallery[edit | edit source]

WikiMD
Navigation: Wellness - Encyclopedia - Health topics - Disease Index‏‎ - Drugs - World Directory - Gray's Anatomy - Keto diet - Recipes

Transform your life with W8MD's budget GLP1 injections from $125

W8mdlogo.png
W8MD weight loss doctors team

W8MD offers a medical weight loss program NYC and a clinic to lose weight in Philadelphia. Our W8MD's physician supervised medical weight loss centers in NYC provides expert medical guidance, and offers telemedicine options for convenience.

Why choose W8MD?

Book Your Appointment

Start your NYC weight loss journey today at our NYC medical weight loss, and Philadelphia medical weight loss Call (718)946-5500 for NY and 215 676 2334 for PA

Search WikiMD

Ad.Tired of being Overweight? Try W8MD's NYC physician weight loss.
Semaglutide (Ozempic / Wegovy and Tirzepatide (Mounjaro / Zepbound) available. Call 718 946 5500.
Advertise on WikiMD

WikiMD's Wellness Encyclopedia

Let Food Be Thy Medicine
Medicine Thy Food - Hippocrates

Medical Disclaimer: WikiMD is not a substitute for professional medical advice. The information on WikiMD is provided as an information resource only, may be incorrect, outdated or misleading, and is not to be used or relied on for any diagnostic or treatment purposes. Please consult your health care provider before making any healthcare decisions or for guidance about a specific medical condition. WikiMD expressly disclaims responsibility, and shall have no liability, for any damages, loss, injury, or liability whatsoever suffered as a result of your reliance on the information contained in this site. By visiting this site you agree to the foregoing terms and conditions, which may from time to time be changed or supplemented by WikiMD. If you do not agree to the foregoing terms and conditions, you should not enter or use this site. See full disclaimer.
Credits:Most images are courtesy of Wikimedia commons, and templates, categories Wikipedia, licensed under CC BY SA or similar.

Contributors: Prab R. Tumpati, MD