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Iwerlipse: Difference between revisions
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(Explain basic mechanism of Arctan Straining and how it constructs the Iwerlipse. Show Iwer's original proof, and equations for the Iwerlipse. Explain the proper history behind the discoveries.) |
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The '''Iwerlipse''' is the range of positions Mario can occupy while he is in the air and cannot turn, assuming that drag is constant. The boundary of the Iwerlipse is reached by '''Arctan Straining''' - the optimal way of air straining that gives the greatest distance along a chosen angle <math>\omega</math> for a given number of frames <math>T</math>. For a single frame <math>T=1</math>, the Iwerlipse is an ellipse with half-width 10 (in the sideways direction) and half-height 1.5 (in the forwards direction). For general <math>T</math>, it resembles an ellipse with half-width <math>10T</math> and half-height <math>3T(T+1)/4</math>, but bulges at the corners, covering more area than an ellipse with the same dimensions. As shown by ''Grassdigger'', the shape is well approximated by a generalised superellipse, and thus ''Pannenkoek2012'' proposed calling this shape the Iwerlipse | The '''Iwerlipse''' (pronounced ee-ver-lipse /ˈivɜlɪps/) is the range of positions Mario can occupy while he is in the air and cannot turn, assuming that drag is constant. The boundary of the Iwerlipse is reached by '''Arctan Straining''' - the optimal way of air straining that gives the greatest distance along a chosen angle <math>\omega</math> for a given number of frames <math>T</math>. For a single frame <math>T=1</math>, the Iwerlipse is an ellipse with half-width 10 (in the sideways direction) and half-height 1.5 (in the forwards direction). For general <math>T</math>, it resembles an ellipse with half-width <math>10T</math> and half-height <math>3T(T+1)/4</math>, but bulges at the corners, covering more area than an ellipse with the same dimensions. As shown by ''Grassdigger'', the shape is well approximated by a generalised superellipse, and thus ''Pannenkoek2012'' proposed calling this shape the Iwerlipse - a pun on the name of ''Iwer Sonsch'', who first discoverered that Arctan Straining was optimal in April 2018. | ||
==Basics== | ==Basics== | ||