TEASER - 02
MAP ENLARGEMENT
If a map (A), having a size equal to A4 paper and scale of 1:100 is enlarged by 200% (2 times) (B), what will be the scale of enlarged map?
A: MAP OF AN AREA WITH SCALE
B: MAP SHOWN AT ‘A’ ABOVE IS ENLARGED
THE EXPLANATION:
We know that a scale or ratio of 1:100 means 1m on map equal to 100 m on ground.
When a map is enlarged or compressed, two things change, (i) scale of map (ii) size or dimension (length & width) of map. Both are linked in such a way that size is inversely proportional to scale (ratio). Larger the map narrower the ratio of a unit distance on map to unit distance on ground.
A map is a kind of a scaled down model of an area (2D) where an actual surface area is represented on a paper (media) suitable for human use. Three types of similar relationship (between actual and model) can be envisaged;
1. Line (length - one dimensional)
2. Area (length & width - two dimensional)
3. Volume (length, width& depth - three dimensional)
1) If a scaled down model of an actual line (one dimensional), having a ratio of 1:100, is enlarged by 200% (2 times) in same way as a map is enlarged, the ratio of model (enlarged) to that of actual one would become 1:50 (exactly half).
ENLARGEMENT OF A LINE (1D)
2) If a map (scale down model) of an area (earth surface), having a scale of 1:100 is enlarged by 200% (2 times) the scale of enlarged map would become 1:71 (approx) and not half as in case of a line.
ENLARGEMENT OF AN AREA (2D)
3) Now if we want to change the scale of same map from 1:100 to 1:50 (half), the map shall have to be enlarged by 400% (4 times) and not 200% (2 times) as would be in case of one dimensional objects such as a line. The area being a 2 dimensional entity, enlargement will have to be done twice along X and Y direction (length and width).
CHANGING SCALE OF AN AREA (2D)
4) On same analogy, if a scale down model (similar to map) of a 3D solid object or volume (having a ratio / scale of 1:100) is enlarged 200% the ratio would be 1:79 and not 1:50 as in case of a line and neither 1:71 as in the case of an area.
ENLARGEMENT OF A SCALE MODEL (SOLID: 3D)
5) Now if scale of same 3D model is changed from 1:100 to 1:50, it would be enlarged by 800% (8 times) and not 400% (4 times) as would be in case of a map. The solid model being 3 dimensional, the enlargement will have to be done volume wise, along X, Y and Z axis.
CHANGING SCALE OF 3D MODEL (VOLUME: 3D)
The solid objects have two elements which can be compared for map to ground type relationship. One is the surface of the object, similar to earths surface (2D) & second is volume contained within its body (3D). Therefore the ratio or scale would depend upon which aspect is being considered while doing enlargement. For example the ratio of surface area would be governed by 2D conversion like that of map to ground, where as ratio for volume would be governed by 3D conversion formula.
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