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In the plane, two figures are called congruent exactly if one can be transformed into the other by translation, rotation, and reflection. What if reflection is excluded, that is, preservation of orientation is required? Is there a term for the resulting equivalence relation?

Wolfgang Jeltsch
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  • I don't think there's a special name for this. "Orientation-preserving isometry" is probably the most common name for such a map. – Sam Hopkins Nov 23 '21 at 23:49
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    I think the question was about the figures, not the maps. Although it is clunky, I think it would be immediately clear what you meant if you said that two such figures are "orientation-preserving isometric"; but, anyway, whatever term you use, you should define it! – LSpice Nov 24 '21 at 00:22
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    To be honest, the notion that reflections would be included in legit "congruence" had slipped my mind, though, now that I think of it, that is consistent with the introductory Euclidean geometry stuff about triangles. As @LSpice says, in this and other situations, it's surely better to say what one means, rather than rely upon volatile terminology... – paul garrett Nov 24 '21 at 02:08
  • Wiktionary suggests superposable as opposed to enantiomorph. – Francois Ziegler Nov 24 '21 at 09:09

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While perhaps not widespread, the term “direct congruence” is used for this equivalence relation.

Wolfgang Jeltsch
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    I hope you won't mind my repeating my earlier comment that, whatever term you use, unless it is absolutely standard in your field, you should define it. – LSpice Nov 24 '21 at 02:04
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    I've never heard "direct congruence" ... I'd not think that this would be a reliable term for communication... – paul garrett Nov 24 '21 at 02:09
  • I didn’t say I wouldn’t want to define this term. The question was just what term to use. I didn’t want to invent a new term if there was already some term in use. – Wolfgang Jeltsch Nov 27 '21 at 00:57