28

포인트를 중심으로 객체를 이동하고 싶을 때 :

    point.x *= cosf(timer.timeElapsed);
    point.y *= sinf(timer.timeElapsed);

8 또는 무한대 부호 궤적에서 점을 이동시키는 방법은 무엇입니까?

mathematics movement trajectory

— Yevhen
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24

Some possibilities:

Lemniscate of Bernoulli

Lemniscate of Gerono

Lemniscate of Booth

Watt's curve

— Marton
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2

Answers should be self-contained, external links may die some day and would render this answer useless. You should quote the important bits from the links you've provided.

— Brian H.

61

As Marton notes, there are several "figure of eight" curves that might fit your needs. Perhaps the simplest is the lemniscate of Gerono, which has the parametrization:

x = cos(t);
y = sin(2*t) / 2;

and looks like this:

Lemniscate of Gerono animation

However, the lemniscate of Bernoulli may be visually more pleasing; it has a parametrization very similar to the lemniscate of Gerono, except that both axes are scaled by a factor of 1/(sin(t)^2 + 1) = 2/(3 - cos(2*t)):

scale = 2 / (3 - cos(2*t));
x = scale * cos(t);
y = scale * sin(2*t) / 2;

It looks like this:

Lemniscate of Bernoulli animation

(Animations made with Maple 13, compressed with GIFsicle.)

— Ilmari Karonen
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Thank you, everybody, for your support, which has earned me my first gold badge here on gamedev! :-)

— Ilmari Karonen

1

+1 for not only posting the links, but also the formulas and graphics ( with sources ).

— rootlocus

2

As is, this should be the accepted answer.

— Brian H.

-1

I randomly found another one using this formula:

x^{2} = y^{2} + 0.1 x^{2.8}

$x^2 = y^2 + 0.1x^{2.8}$

As plotted by Wolfram Alpha:

half of an infinity symbol

— user75095
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Unlike the other answers, this one is currently not presented in parametric form that lets us easily step the position forward over time t. I'd recommend including a description of how you would use this formula to position a moving object over time.

— DMGregory

-4

$((x + 1)^{2} + y^{2}) ((x - 1)^{2} + y^{2}) = 1$ $((x+1)^2+y^2)((x-1)^2+y^2)=1$

half of an infinity symbol

The product of the distances from any point on that curve to (-1, 0) and to (1,0) is constant and equals to 1.

— user111508
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4

This answer provides a formula modelling such a curve, but not a method to "move an object" in such a way that it follows that curve. Please consider elaborating on the answer to indicate how you would use this math to move an object in a game.

— DMGregory

"무한대"또는 "그림 8"궤적에서 개체를 어떻게 이동합니까?

((x+1)2+y2)((x−1)2+y2)=1((x+1)2+y2)((x−1)2+y2)=1((x+1)^2+y^2)((x-1)^2+y^2)=1

The product of the distances from any point on that curve to (-1, 0) and to (1,0) is constant and equals to 1.

$((x + 1)^{2} + y^{2}) ((x - 1)^{2} + y^{2}) = 1$ $((x+1)^2+y^2)((x-1)^2+y^2)=1$