Similarity between percepts and concepts is used to accomplish many everyday tasks, e.g., object identification; so this similarity is widely used to construct geometrical spaces that represent stimulus qualities, but the intrinsic validity of the geometry, i.e., whether similarity operations support a particular geometry, is almost never tested critically. We introduce an experimental approach for equating relative similarities by setting perceived midpoints between pairs of stimuli. Midpoint settings are used with Varignon's Theorem to test the intrinsic geometry of a representation space, and its mapping to a physical space of stimuli. For perceptual color space, we demonstrate that geometrical structure depends on the mental representation used in judging similarity: An affine geometry was valid when observers used an opponent-color mental representation. Similarities based on a conceptual space of complementary colors thus power a geometric coordinate system. An affine geometry implies that similarity can be judged within straight lines and across parallel lines, and its neural coding could involve ratios of responses. We show that this perceptual space is invariant to changes in illumination color, providing a formal justification to generalize color constancy results measured for color categories, to all of color space. The midpoint measurements deviate significantly from midpoints in the extensively used “uniform” color spaces CIELAB and CIELUV, showing that these spaces do not provide adequate metric representation of perceived colors. Our paradigm can thus test for intrinsic geometrical assumptions underlying the representation space for many perceptual modalities, and for the extrinsic perceptual geometry of the space of physical stimuli.
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