Tuesday, March 29, 2011

Scale models

I read the xkcd comics regularly, but I rarely venture into the forums. For this comic regarding model railways and towns I did. Not because of the comic per se, but the alt text regarding H0. Simply for a laugh I wondered how quickly the thread would degenerate into whether it's H0 or HO; about four comments in :-)

However the third post by beet31425 caught my eye.

By the way, if you did build a layout of your town in your basement, the Brouwer Fixed Point Theorem (http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem) implies that there would be a point in your layout that lies directly over the point in the town it corresponds to.

Similarly: If you're in the US, place a map of the US on the floor. There exists a point on the map lying directly over the point in the US it corresponds to.

My thought was "Why would you need to invoke the Brouwer Fixed Point Theorem at all?" If an exact scale model of an object is placed within the confines of the object it is modelled on there has to be such a corresponding point - it's the scale point.

The scale point is a point such that if all measurements taken from it to the extent of the model were multiplied by the same scale factor it would match and overlap the real object exactly. Used as the origin (0,0) by definition it doesn't move when you multiply all the other points by the scale factor - 0 times anything equals 0.

The scale point can be found either mathematically or geometrically and I've shown the latter method above. Mathematically it can be derived in one dimension (a line) that can simply be repeated for each perpendicular dimension.

To use a different point, as shown below, the point in the model corresponds to a point in the object where the line intersects the model. The distance from the model corner to the intersection corresponds to the same distance from the object corner multiplied by the scale factor. The distance from the intersection to the point is likewise multiplied and to keep it simple we can use the same angle (we could draw two lines and use them instead just to prove the point)

Although other points can exist within both the model and object if we used them as our zero (scale) point and scaled up our newly sized model wouldn't exactly overlap the object, it would be a little (or a lot) too far over. Only at the scale point can we multiply all the distances by the same amount and match reality exactly. The consequence otherwise are shown below using the yellow point and multiplying  by the scale factor

Mathematically the straight line offset is equal to the distance between the two points multiplied by the scale factor minus one.

If someone builds a scale model of the town in their basement, by definition the scale point for that model lies within the basement of that model house. No fancy mathematics required.