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Is it called

Auf dem transformierten Bild

or

Auf dem transformiertem Bild

and whats the rule for this?

Edit: All found answers are not covering dative.

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    The question is about the same grammatical phenomenon (weak vs strong declension of adjectives), but the dative isn't covered. – RHa Aug 12 '17 at 18:55
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    The adjective ending on n is correct here. Loosely spoken, there is already an m in dem, so you don't need another. So it is auf dem transformierten Bild, or auf einem transformierten Bild, but auf transformiertem Bild. I don't know the exact grammatical rule, but the only one m "rule" has always worked for me. <g> – Rudy Velthuis Aug 12 '17 at 19:17
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The rule is one m and one n, basically. If the article ends with an m, the subject will end with an n. That's how I learnt it, at least. Just so know, most native speakers will not know / do this wrong. ;)

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    Auf dem transformierten, roten, scharfen, kleinen Bild - one m, multiple n. – user unknown Aug 12 '17 at 22:35
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Since there is a definite article, the weak declension is used for the adjective, so transformierten (dative) is correct.

Since it's das Bild, the gender is neuter. Here's the forms for all cases:

+----+---------------------+--------------------+
|Case|Singular             |Plural              |
+----+---------------------+--------------------+
|Nom |das transformierte   |die transformierten |
|Gen |des transformierten  |der transformierten |
|Dat |dem transformierten  |den transformierten |
|Akk |das transformierte   |die transformierten |
+----+---------------------+--------------------+

Without an article, the strong declension is used:

+----+-----------------+----------------+
|Case|Singular         |Plural          |
+----+-----------------+----------------+
|Nom |transformiertes  |transformierte  |
|Gen |transformierten  |transformierter |
|Dat |transformiertem  |transformierten |
|Akk |transformiertes  |transformierte  |
+----+-----------------+----------------+

So if the "dem" were missing, transformiertem would be correct.

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