In Spherical Mercator Tile coordinates, how to go one zoom-level deeper but span the same area?

Question

For an arbitrary tile region defined by (z1, x1, y1) in Standard Web Mercator Tile format (Spherical Mercator) , how can we increase z1 to z1 + 1 but span the exact area defined by (z1, x1, y1)?

For example, consider t1=(2,2,1). If we want zoom=3 for the same region, what (x,y) coordinates should we fetch and glue together?

Here is what I tried:

• target=(latitude, longitude), zoom = z1
• convert target to (x1, y1)
• fetch tile: t1 = (z1, x1, y1) → (desired target area)
• z2 = z1 + 1 → (desired zoom-level (detail level))
• convert target in (lat/long) with z2 to get (x2,y2): t2 = (z2, x2, y2)
• fetch 8 tiles in the neighborhood of t2, (like (z2, x2 ± 1, y2 ± 1))
• assemble these 9 tiles together:

n1-n2-n3
n4-t2-n5
n6-n7-n8

But this solution does NOT span the same area as t1
Example: target=Berlin, z1=2 (one tile), z2=3 (nine tiles)

Note: I asked the same question here, but due to the Mathematical nature of the question, I re-posted here to get more mathematical/algorithmic (rather than practical/programming) answers.

Answer 1

I hope I understood the question correctly.

From the description in the link you provided, it is clear that we have four tiles in the higher zoom level representing the same area as one tile in the original zoom level, not nine tiles. The tile $(x,y)$ in zoom level $z$ shows the same area as the tiles $$ (2x\;\;\;\;\;\;,2y\;\;\;\;\;\;) \\ (2x+1,2y\;\;\;\;\;\;) \\ (2x\;\;\;\;\;\;,2y+1) \\ (2x+1,2y+1) $$ in zoom level $z+1.$