{"id":111,"date":"2026-03-24T11:10:21","date_gmt":"2026-03-24T09:10:21","guid":{"rendered":"https:\/\/blogs.uef.fi\/photogrammetry\/?page_id=111"},"modified":"2026-03-24T11:10:22","modified_gmt":"2026-03-24T09:10:22","slug":"2d-conformal-transformation","status":"publish","type":"page","link":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/","title":{"rendered":"2D Conformal Transformation"},"content":{"rendered":"\n


In photogrammetry, we often need to map points from one coordinate system to another. One of the most important and widely used transformations for this purpose is the 2D conformal transformation<\/strong>.
This transformation preserves the shape and angles<\/strong> of objects while allowing for scaling, rotation, and translation.

\ud83d\udccc What Is a 2D Conformal Transformation?
A 2D conformal transformation connects two Cartesian coordinate systems using:
Scale (\u03bb)<\/strong>
Rotation (\u03b8)<\/strong>
Translation (x\u2080, y\u2080)<\/strong>
It is commonly used when two coordinate systems differ only by orientation, position, and size\u2014but not by distortion.

\ud83d\udd22 Mathematical Formulation
The transformation is defined as:
x\u2032 = \u03bb \u00b7 cos(\u03b8) \u00b7 x \u2212 \u03bb \u00b7 sin(\u03b8) \u00b7 y + x\u2080
y\u2032 = \u03bb \u00b7 sin(\u03b8) \u00b7 x + \u03bb \u00b7 cos(\u03b8) \u00b7 y + y\u2080
Where:
(x, y) \u2192 coordinates in the original system
(x\u2032, y\u2032) \u2192 coordinates in the target system
\u03bb \u2192 scale factor
\u03b8 \u2192 rotation angle
(x\u2080, y\u2080) \u2192 translation (shift between systems)

\ud83e\udde9 Simplified Form (For Computation)
To make calculations easier, we define new variables:
a = \u03bb \u00b7 cos(\u03b8)
b = \u03bb \u00b7 sin(\u03b8)
This simplifies the equations to:
x\u2032 = a\u00b7x \u2212 b\u00b7y + c
y\u2032 = b\u00b7x + a\u00b7y + d
Where:
c = x\u2080
d = y\u2080
This form is much more convenient for numerical solutions.

\ud83e\uddee Solving the Transformation
To compute the unknown parameters (a, b, c, d), we use corresponding points<\/strong> from both coordinate systems.
For N points<\/strong>, we build a system of equations:
Each point gives 2 equations
Total equations = 2N
This system can be written in matrix form:
L = A \u00b7 X<\/strong>
Where:
L<\/strong> \u2192 observed coordinates (target system)
A<\/strong> \u2192 coefficient matrix
X<\/strong> \u2192 unknown parameters

\ud83d\udcca Least Squares Solution
Since real data often contains noise, we solve the system using the least squares method<\/strong>:
X\u0302 = (A\u1d40A)\u207b\u00b9 A\u1d40L<\/strong>
This gives the best-fit parameters that minimize error.
After solving, we can compute residuals:
V\u0302 = A \u00b7 X\u0302 \u2212 L<\/strong>
These residuals help evaluate the accuracy of the transformation.

\u2699\ufe0f Degrees of Freedom
The 2D conformal transformation has 4 unknown parameters<\/strong>:
a, b \u2192 rotation + scale
c, d \u2192 translation
This means:
Minimum required points = 2 points<\/strong> (but more are recommended for accuracy)

\ud83d\udccd When to Use It
Use 2D conformal transformation when:
Shapes must remain unchanged (no distortion)
Angles must be preserved
Only rotation, scaling, and shifting are needed
Typical applications:
Image alignment
Map registration
Coordinate system conversion

\u26a0\ufe0f Limitations
Cannot handle skew or non-uniform scaling
Not suitable when axes are not perpendicular
Cannot model perspective effects
For more complex cases, use:
Affine transformation
Projective transformation

\ud83d\ude80 Summary
Preserves shape and angles<\/strong>
Uses scale, rotation, and translation<\/strong>
Simplified using secondary variables (a, b,<\/p>\n","protected":false},"excerpt":{"rendered":"

In photogrammetry, we often need to map points from one coordinate system to another. One of the most important and widely used transformations for this purpose is the 2D conformal transformation.This transformation preserves the shape and angles of objects while allowing for scaling, rotation, and translation. \ud83d\udccc What Is a 2D Conformal Transformation?A 2D conformal […]<\/p>\n","protected":false},"author":746,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-111","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\n2D Conformal Transformation - Learn Photogrammetry<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"2D Conformal Transformation - Learn Photogrammetry\" \/>\n<meta property=\"og:description\" content=\"In photogrammetry, we often need to map points from one coordinate system to another. 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One of the most important and widely used transformations for this purpose is the 2D conformal transformation.This transformation preserves the shape and angles of objects while allowing for scaling, rotation, and translation. \ud83d\udccc What Is a 2D Conformal Transformation?A 2D conformal […]","og_url":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/","og_site_name":"Learn Photogrammetry","article_modified_time":"2026-03-24T09:10:22+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/","url":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/","name":"2D Conformal Transformation - Learn Photogrammetry","isPartOf":{"@id":"https:\/\/blogs.uef.fi\/photogrammetry\/#website"},"datePublished":"2026-03-24T09:10:21+00:00","dateModified":"2026-03-24T09:10:22+00:00","breadcrumb":{"@id":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/blogs.uef.fi\/photogrammetry\/2d-conformal-transformation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/blogs.uef.fi\/photogrammetry\/"},{"@type":"ListItem","position":2,"name":"2D Conformal Transformation"}]},{"@type":"WebSite","@id":"https:\/\/blogs.uef.fi\/photogrammetry\/#website","url":"https:\/\/blogs.uef.fi\/photogrammetry\/","name":"Learn Photogrammetry","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/blogs.uef.fi\/photogrammetry\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"}]}},"_links":{"self":[{"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/pages\/111","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/users\/746"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/comments?post=111"}],"version-history":[{"count":1,"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/pages\/111\/revisions"}],"predecessor-version":[{"id":113,"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/pages\/111\/revisions\/113"}],"wp:attachment":[{"href":"https:\/\/blogs.uef.fi\/photogrammetry\/wp-json\/wp\/v2\/media?parent=111"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}