577
edits
Iwerlipse: Difference between revisions
removed discord media reference for in-wiki ref
(Math formatting) |
(removed discord media reference for in-wiki ref) |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
The Iwerlipse is the range of positions Mario can occupy when Mario is in the air and cannot turn, assuming the joystick input polling were continuous. It is defined as an area bounded by a shape drawn by the solution to optimizing a set of 2 differential equations. Due to the shape being similar to a superellipse (after Grassdigger brute forced for the solution in Mathematica), Pannenkoek2012 proposed calling this shape the "Iwerlipse" on August 2018 (pronounced eeverlipse) as a pun on the name of Iwer Sonsch, one of the people involved in this problem. | The '''Iwerlipse''' is the range of positions Mario can occupy when Mario is in the air and cannot turn, assuming the joystick input polling were continuous. It is defined as an area bounded by a shape drawn by the solution to optimizing a set of 2 differential equations. Due to the shape being similar to a superellipse (after Grassdigger brute forced for the solution in Mathematica), Pannenkoek2012 proposed calling this shape the "Iwerlipse" on August 2018 (pronounced eeverlipse /ˈivɜlɪps/) as a pun on the name of Iwer Sonsch, one of the people involved in this problem. | ||
| Line 21: | Line 21: | ||
Iwer stated on April 2018 that optimal straining for the Air_no_Turn action is <math>\text{dyaw}(t,r) = \text{acotan}(0.15r*t) </math>. | Iwer stated on April 2018 that optimal straining for the Air_no_Turn action is <math>\text{dyaw}(t,r) = \text{acotan}(0.15r*t) </math>. | ||
On October 2019, | On October 2019, trex31415 provided a full analytical proof for Iwer's claim, using a generalized 2nd-order Euler-Lagrange equation. | ||
By plugging Iwer and Trex's equation to solve for x and y, we obtain the parametric equations, the solution to the Iwerlipse: | By plugging Iwer and Trex's equation to solve for x and y, we obtain the parametric equations, the solution to the Iwerlipse after <math>n</math> frames: | ||
<math>x = \pm \frac{v_xnt}{\sinh(t)} </math> | <math>x(n,t) = \pm \frac{v_xnt}{\sinh(t)} </math> | ||
<math>y = v_in + \frac{a}{2}n^2\bigg(\coth(t)-\frac{t}{\sinh^2(t)}-\frac{D}{a}\bigg)</math> | <math>y(n,t) = v_in + \frac{a}{2}n^2\bigg(\coth(t)-\frac{t}{\sinh^2(t)}-\frac{D}{a}\bigg)</math> | ||
==See Also== | |||
[[Straining]] | |||
==References== | |||
Proof of the optimal straining equation :<br> | |||
[[File:IwerlipseProof.png]] | |||
[[Category:Mechanics]] | |||