Purpose:

Another important teleport technique are No Impact Lands (NILs). A NIL occurs when Mewtwo’s downward velocity is low enough where there is no landing lag. NILs occur at latter frames in teleport where position calculations become more complicated

This model will be able to give Mewtwo’s position given that the frame, control stick angle at frame 5, and control stick radius at frame 5 are given.

Conic Sections

A cross-section of a cone can result in 4 different types of 2-D shapes:

• Circle
• Ellipse
• Parabola
• Hyperbola

Circle

Equation of a Circle is represented in Cartesian Coordinates here:

( x – h )^2 + ( y – k )^2 = r^2

where r = the radius of the circle, h = x-coordinate of the origin, and k = y-coordinate of the origin

Ellipse

Equation of an Ellipse is represented in Cartesian Coordinates here:

( x – h )^2 / a^2 + ( y – k )^2 / b^2 = 1

where a = semi major axis or the position that Mewtwo travels with a horizontal teleport and b = semi minor axis or the position that Mewtwo travels with a vertical teleport.

Frame 17 Model

Circle Model:

r = Mewtwo’s Position = 20.36 * R1 + 30.54

Ellipse Model:

a = ( 2 * R1 + 3 ) * ( 10 + 0.2 * n ) – 0.016 * n * (n + 1 ) * 0.5 * sec( E )

b = ( 2 * R1 + 3 ) * ( 10 + 0.18 * ( ( 1 – 0.9^n ) / ( 1 – 0.9 ) ) )

where E = the angle made with the control stick at frame 5, R1 = the radius made with the control stick at frame 5, and n = 1 (at frame 17)

Why is the ellipse model more complicated?

The ellipse model more accurately describes position at frame 17 but because a ~= b = r, we are able to use the circle model and still get accurate data. This assumption does not work at latter frames.

How do we get the x and y coordinate of Mewtwo’s position at a certain frame?

Mewtwo’s X Position = a * cos ( E )

Mewtwo’s Y Position = b * sin ( E )

Equation for each frame

f = current frame

Frame 7 to 16:

v = f – 6

r = ( 2 * R1 + 3 ) * v

Frame 17 to 23:

v = 16 – 6

n = f – 16

a = ( 2 * R1 + 3 ) * ( v + 0.2 * n ) – 0.016 * n * ( n + 1 ) * 0.5 * sec( E )

b = ( 2 * R1 + 3 ) * ( v + 0.18 * ( ( 1 – 0.9^n ) / ( 1 – 0.9 ) ) )

Frame 24 to 31:

n = 23 – 16

m = f – 24

a = ( 2 * R1 + 3 ) * ( v + 0.2 * n + 0.2 * m ) – 0.016 * ( n * ( n + 1 ) * 0.5 + 8 * m ) * sec( E )

b = ( 2 * R1 + 3 ) * ( v + 0.18 * ( ( 1 – 0.9^n ) / ( 1 – 0.9 ) ) + 0.18 * m * 0.9^n )

Frame 32+:

m = 31 – 24

l = f – 31

a = ( 2 * R1 + 3 ) * ( v + 0.2 * n + 0.2 * m + 0.2 * l ) – 0.016 * ( n * ( n + 1 ) * 0.5 + 8 * m + 8 * l + l * ( l + 1 ) * 0.5 ) * sec( E )

b = “”

k = 0.082 * 0.5 * m * ( m + 1 )

Limits of the Ellipse Model

This model falls apart when Mewtwo reaches it’s maximum downward velocity of -1.5. This is a problem because this does not occur all at the same frame and therefore is difficult to represent in the semi major and minor axis’.

Another problem is that the model is not represented by one ellipse at frame 17+ but by two different shifted ellipses. The current ‘a’ is sufficient but does not fit the status of a conic section when it is used as the semi major axis.

Conclusions

With this information, Mewtwo’s position during teleport, until a downward velocity of -1.5 is reached, should be calculable.

References

1. By http://commons.wikimedia.org/wiki/User:Magister_Mathematicae – File:Secciones_cónicas.svg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18556148
2. By Ellipse_parameters_1_en.svg: Moranaderivative work: Bomazi (talk) – Ellipse_parameters_1_en.svg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=14648413
3. By CheCheDaWaff – This file was derived from: Eccentric and true anomaly.PNG:, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=48384905