Curve Movement

[Ayanami]

I've been pondering on creating a curve movement. Please note that this isn't a jump spell and everything is on the 2D plane. So, I've been trying to create a basic parabola (quadratic) curve. The only information I have is the starting point, end point and the maximum point. Is there a way to create this curve movement with only this 3 points?

 

[kingkingyyk3]

You can try to form some quadratic equation and test it out.

 

[Endless_Trev]

Just for simplicity I would say you should use the local maxima or minima as the centre of the of the parabola's x and y axis. Also try to come up with a standard form for your parabola, say y=x^2/8 or something, depending on what curvature you want to achieve. Since you know the general equation the derivative would be x/4, so you would just have to sub in the x value of a co-ordinate into x/4 and that would give you the gradient of the tangent at that line, so you can move the unit in the direction of the tangent. Of course this is all using the maxima or minima as (0,0), meaning you will always have to treat the differing positions in relation to the origin of the graph you are basing around the parabola. Ive tried myself using things like leap, but I get lazy and revert to primitive versions of creating parabolic motion.

Yes it should be relatively easy if you know what you're doing, which I assume you do.

 

[uberfoop]

Bleh, he's not asking how to move something along a curve, he's asking how to interpolate using 3 points.

If you want the formula in terms of x and y, of the form y=ax^2+bx+c, and one of your points is the parabolic vertex, you can only really use 2 points, since parabolas of that form are symmetric. In this case, you can use:
Y=AX^2+AB(B-2X)+C
(This is a parabolic form such that B is the vertex's x, C is the vertex's y, and A is what it is in standard parabolic form)
Using this formula, if the vertex is (U,V) and the other point is (J,K), then:
A=(K-V)/(J-U)^2
B=U
C=V



However, this sort of math is kind of limited. If you're interested in interpolating 3 points along a second-degree curve, for many purposes Bezier Curves work very well.
They can be applied to any number of dimensions with little pain (note how wiki gives the formula for a single dimension, with coordinate components given as 'P0', 'P1', and 'P2'), are neato in that they're always parameterized by t from t=0 to t=1, and they have the interesting property that their gradient approaches the slope from the center point to each endpoint at the respective endpoints.

 

[Never_Quit]

parametric would be easiest when dealing with unit position.

y = (t^2)(height constant)
x = (t+1)(distance constant)

where t is the phase of the movement.

EDIT: The higher number of loops for t that you run, the smoother the animation. The smother then animation, the lower the distant and height constant must be relative to the same path of movement. Pretty basic really. You would make the unit instantaneously move to a point (unit's starting position) with offset: x and y.

 

[Ayanami]

Bleh, he's not asking how to move something along a curve, he's asking how to interpolate using 3 points.

If you want the formula in terms of x and y, of the form y=ax^2+bx+c, and one of your points is the parabolic vertex, you can only really use 2 points, since parabolas of that form are symmetric. In this case, you can use:
Y=AX^2+AB(B-2X)+C
(This is a parabolic form such that B is the vertex's x, C is the vertex's y, and A is what it is in standard parabolic form)
Using this formula, if the vertex is (U,V) and the other point is (J,K), then:
A=(K-V)/(J-U)^2
B=U
C=V

However, this sort of math is kind of limited. If you're interested in interpolating 3 points along a second-degree curve, for many purposes Bezier Curves work very well.
They can be applied to any number of dimensions with little pain (note how wiki gives the formula for a single dimension, with coordinate components given as 'P0', 'P1', and 'P2'), are neato in that they're always parameterized by t from t=0 to t=1, and they have the interesting property that their gradient approaches the slope from the center point to each endpoint at the respective endpoints.

Would you mind elaborating more on Bezier Curves?

 

[13lade619]

may I ask what level of education you're in now? (Middle school / high school / college).

i'll try to elaborate... well yeah, if i can remember how to convert parametric to cartesian which i learned last semester.. *EDIT*

Basically, what you need is the Quadratic Bezier Curve.

Equation

you'll need the points P0, P1 and P2. and find the values from t=0 to t=1.
it's neat that it's in parametric so you can use them in x-y coords or even x-y-z.

 

[Never_Quit]

may I ask what level of education you're in now? (Middle school / high school / college).

i'll try to elaborate... well yeah, if i can remember how to convert parametric to cartesian which i learned last semester.. *EDIT*

Basically, what you need is the Quadratic Bezier Curve.

Equation

you'll need the points P0, P1 and P2. and find the values from t=0 to t=1.
it's neat that it's in parametric so you can use them in x-y coords or even x-y-z.

so if I understand correctly, using this function, for a loop in code, you would have t equal 1/a where:
for each int a from DEGREE_OF_ACCURACY to 1

the loop would run from the smallest values to the largest values.

 

[13lade619]

a loop in code

loops arent enough, you need timers for more executions per second.

 

[Ayanami]

may I ask what level of education you're in now? (Middle school / high school / college).

i'll try to elaborate... well yeah, if i can remember how to convert parametric to cartesian which i learned last semester.. *EDIT*

Basically, what you need is the Quadratic Bezier Curve.

Equation

you'll need the points P0, P1 and P2. and find the values from t=0 to t=1.
it's neat that it's in parametric so you can use them in x-y coords or even x-y-z.

From this, I can see that the curve does not pass through P1 right? Is there a solution if I require the curve to pass through all 3 points?

 

[13lade619]

i think you cant.. why do you want such precision?

i managed to use the function to create effects in a loop, i'm sure you can use it for unit movement.

function Trig_Untitled_Trigger_001_Actions takes nothing returns nothing
    local unit u = gg_unit_Hpal_0000
    
    //origin
    local real p0x = GetUnitX(u)
    local real p0y = GetUnitY(u)
    
    //outer point
    local real p1x = p0x + 700*CosBJ(GetUnitFacing(u)+80)
    local real p1y = p0y + 700*SinBJ(GetUnitFacing(u)+80)
    
    //destination
    local real p2x = p0x + 1000*CosBJ(GetUnitFacing(u))
    local real p2y = p0y + 1000*SinBJ(GetUnitFacing(u))
    
    //parameter
    local real t = 0
    
    
    local real X
    local real Y
    loop
        exitwhen t>1
        
        // B(t) = (1-t)^2 p0 + 2(1-t)t p1 + t^2 p2
        set X = (((1-t)*(1-t))*p0x)+(2*(1-t)*t*p1x)+(t*t*p2x)
        set Y = (((1-t)*(1-t))*p0y)+(2*(1-t)*t*p1y)+(t*t*p2y)
        
        call DestroyEffect(AddSpecialEffect("units\\nightelf\\Wisp\\Wisp.mdl",X,Y))
        
        // Speed / Frequency of Transition
        set t = t+.05
    endloop
    
endfunction

//===========================================================================
function InitTrig_Untitled_Trigger_001 takes nothing returns nothing
    set gg_trg_Untitled_Trigger_001 = CreateTrigger(  )
    call TriggerRegisterPlayerKeyEventBJ( gg_trg_Untitled_Trigger_001, Player(0), bj_KEYEVENTTYPE_DEPRESS, bj_KEYEVENTKEY_LEFT )
    call TriggerAddAction( gg_trg_Untitled_Trigger_001, function Trig_Untitled_Trigger_001_Actions )
endfunction

 

[Never_Quit]

From this, I can see that the curve does not pass through P1 right? Is there a solution if I require the curve to pass through all 3 points?

You will need to use a parametric equation with distance constants for that.
You solve for your distance constants by putting in the max variables you want.

Btw ^^^ pic up there is neat

 

[SineCosine]

Damn..
I so have to bookmark this page.

 

[uberfoop]

From this, I can see that the curve does not pass through P1 right? Is there a solution if I require the curve to pass through all 3 points?

Sure. Use 2 Bezier curves to go through all 3 points. To make sure the spline stays smooth through the middle point, the midpoints of each curve will have to be generated points equidistant and at opposite angles from the middle point, like so, where the black points are the 3 points being interpolated and the maroon points are the "generated" midpoints of each curve:

How you set up those midpoints is arbitrary, but for the curve to be entirely smooth (smoothness to the first degree, that is. Maintaining both curvature smoothness AND directional smoothness on second-degree Bezier splines of arbitrary shape is impossible, but you shouldn't worry about that unless you're running a camera eye tangent to the curve), the two red points must:
1) Be the SAME distance from the middle point.
2) Be extrapolated from the middle point at opposite angles.


Since this leaves much stuff arbitrary, here is an example of some guidelines which should create OK-looking curves under most circumstances:
Let's call the 3 points being interpolated points 1, 3, and 5. The red points will be points 2 and 4. Let's call the angle from point 1 to point 5 "T" and the distance between points 1 and 5 "D". Point 4 will be projected from point 3 by distance D/4 in direction T. Point 2 will be projected from point 3 by distance D/4 in direction T+180.

Damn..
I so have to bookmark this page.

Hardly. So far this is a relatively boring interpolation discussion. If we want to make it more exciting, we can bring up stuff like polynomial interpolation. EG:

Given points (X0,Y0),(X1,Y1),(X2,Y2), to interpolate them on a curve of form Y=Ax^2+Bx+C, you do:

Anyway, that's why we aren't using straight polynomial interpolation, in case anyone was wondering.

 

[SineCosine]

All..
This..
Math..
Hurts..