# Christmas Tree Light: Investigating the Math Behind Its Curve

• Livethefire
In summary, the conversation discusses the observation of a circle and a hyperbola when shining a light onto a surface. The question is raised about the relationship between the equations for a circle and hyperbola and the substitution of "iy" for y. It is determined that the rotation of the axis by 90 degrees changes the view from a circle to a hyperbola, but the relevance of using "i" as a 90 degree operator is questioned. The conversation also explores the concept of a cone and how it intersects with a plane to create an ellipse or hyperbola. However, it is unclear how "i" plays a role in this equation.
Livethefire
Forgive the sloppy use of math and inability to produce an image. I noticed this last christmas.

If you have a fairy light ( or perhaps any LED etc), and shine it normal to a surface, you see a circle. If you place the light flat on the surface you see a curve - to me the fairy lights' curve looks like a hyperbola.

Does this have any relation to the equation of a circle:
$$x^2+y^2=const.$$
And Hyperbola:
$$x^2-y^2=const.$$
And subsitution for 90 degrees rotation? :
$$y\rightarrow iy$$

If so, how does this even work? The experiment is all in real space. If not, is this just sloppy use of math? Any significance?

Thanks

Hi Livethefire!

If the light comes out in a cone,

then the shape will be the intersection of a cone with a plane …

in other words, a conic section

Ah yes!

But is there any relevance or justified motivation to present such a thing by substituting "iy" in a circular equation?

not following you

What I was saying in post #1 was to sub iy for y in the first equation you get the second. In other words, rotating the axis 90 degrees changes the view from a circle to a hyperbola.

Sometimes i is used as a 90 degree operator, yet I think my reasoning is unsound, thus i am asking here for insight.

if the cone has semiangle λ along the z-axis, then its equation is

z2 = (x2 + y2)tan2λ,

so a plane z = xtanθ + c cuts it at x2(tan2λ - tan2θ) - 2cxtanθ + y2tan2λ = c2,

which is an ellipse or hyperbola according to whether λ is greater or less than θ

(but i don't see where i comes into it)

## 1. How do Christmas tree lights create a curved shape?

The curve of Christmas tree lights is created by the concept of "catenary," which is the shape formed by a chain or rope when it is suspended between two points. The lights are hung on the tree in a way that mimics this natural curve, creating a visually pleasing effect.

## 2. What is the mathematical equation behind the curve of Christmas tree lights?

The mathematical equation for a catenary curve is y = a cosh(x/a), where y represents the height of the curve at a given point, x represents the horizontal distance from the center of the curve, and a is a constant related to the weight and length of the chain or rope. This equation can be applied to the curve of Christmas tree lights as well.

## 3. Are there any other factors besides math that contribute to the curve of Christmas tree lights?

Yes, there are other factors that can affect the shape of Christmas tree lights, such as the weight and length of the lights themselves, the type of tree they are hung on, and the spacing between each light. These factors can vary and may result in slightly different curves.

## 4. Can the curve of Christmas tree lights be predicted or controlled?

Yes, the curve of Christmas tree lights can be predicted and controlled to a certain extent. By using the mathematical equation for a catenary curve and considering the other factors mentioned, the shape of the lights can be planned and adjusted to achieve a desired effect.

## 5. Is there a specific reason why Christmas tree lights are hung in a curved shape?

The curved shape of Christmas tree lights is primarily for aesthetic purposes. It creates a more visually appealing and dynamic display compared to straight lines. The curve also helps to evenly distribute the lights on the tree and prevent any gaps or clumps of lights.

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