Why Doesn't r=a(1+cosθ) Have a Dimple?

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The discussion centers on the shape of the curve defined by r = a(1+cosθ) and its classification regarding the presence of a dimple. Participants debate whether the curve's cusp qualifies as a dimple, with some arguing that dimples should have a flat shape, unlike the point observed in the cardioid. The conversation also references a source that categorizes cardioids as lacking dimples. Ultimately, the distinction between a degenerate loop and a dimple is deemed less significant than the overall classification of the curve. The nuances of these definitions highlight the complexity of geometric shapes and their characteristics.
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Why doesn't r = a(1+cosθ) have a dimple? I mean p=1, q=1 so q≤ p<2q and therefore r = a(1+cosθ) should have a dimple (like the curve in the bottom right corner of the image above)?
 
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It does have a dimple with a point. If q gets any smaller it loops inside itself.
 
LCKurtz said:
It does have a dimple with a point. If q gets any smaller it loops inside itself.

Is that considered to be a dimple? I thought dimples have a ''flat'' shape to them like the curve in the box on the bottom right?

Also, according to this page:

http://www.jstor.org/discover/10.2307/3026536?uid=3738032&uid=2&uid=4&sid=21104158779553

A cardioid doesn't have a dimple? (Table 1)

Thanks
 
influx said:
Is that considered to be a dimple? I thought dimples have a ''flat'' shape to them like the curve in the box on the bottom right?

Also, according to this page:

http://www.jstor.org/discover/10.2307/3026536?uid=3738032&uid=2&uid=4&sid=21104158779553

A cardioid doesn't have a dimple? (Table 1)

Thanks
Whether the cusp of a cardioid is regarded as a degenerate loop, a degenerate dimple, or distinct from both, doesn't strike me as terribly important. If I tell you a curve is an ellipse, does that mean it's not a circle?
 

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