Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 adjective Mathematics Designating or specifying a mapping of a surface or region upon another surface so that all angles between intersecting curves remain unchanged.
 adjective Of or relating to a map projection in which small areas are rendered with true shape.
from The Century Dictionary.
 Of the same form.
 In mathematics, conserving angles or establishing infinitesimal similarity.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective Describing something that
conforms , especially that matches the shape of something.  adjective cartography Describing a map
projection which has the property of preserving relative angles over small scales (except at a limited number of distinct points). On such map projections the scale depends only location but not direction. Also referred to asorthomorphic .
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
Examples

You argue that there are two difficulties with AdS/CFT: that strong forms of it are inconsistent with the bending of light by gravitational fields, and that the evidence supports a weaker relation that you call conformal induction.

Mr. Penrose points out that, as one goes back toward the infinite energy density of the Big Bang, the geometry of the early universe could in some sense be called conformal.

Penrose's cyclic cosmological model is a particular application of his conformal compactification construction, which takes any spacetime, whether or not it contains singularities, and whether or not it is infinite in time or space, and constructs a finite (compactified) spacetime with boundary, whose metric is related to the original metric of spacetime by a locally variable scale factor & #937, called the conformal factor.

The corneas were coated in 250 nm of nickel using a technique developed at Penn State called conformalevaporatedfilmbyrotation.

The Mercator is also a "conformal" map projection.

Despite the English way of not really asking questions much in seminars and the like (which drove me nuts when I was here, and still does now), I think that Durham offers the best allround training for students or postdocs in theoretical high energy physics (either phenomenological or more formal, such as conformal field theory, integrable systems, strings) in the UK, and can give several places around the world a run for their money.

I don’t want to comment further on this until I’ve done some more reading, but if you want to see a Penrose diagram (aka conformal diagram) of an evaporating black hole, there’s one on page 413 of Wald’s General Relativity.

Researchers at Wake Forest University's Center for Nanotechnology and Molecular Materials are working to create flexible, or "conformal," organic solar cells that can be wrapped around surfaces, rolled up or even painted onto structures.

The theories on the worldsheet must be conformal which is string theory's way to demand the spacetime equations of motion  the betafunctions are the equations of motion.

Researchers at Wake Forest University's Center for Nanotechnology and Molecular Materials are working to create flexible, or "conformal," organic solar cells that can be wrapped around surfaces, rolled up or even painted onto structures.
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