How to scale a point relative to a specific point as origin, instead of origin (0,0)?

Question

I have the scaling factors sf=[0.5,0.75,0.85,1,1.25,1.5,1.75,2] , and I want to calculate the coordinates of the point e=[70, 140] (blue line) by scaling relative to the center_point=[89, 121] (red point in the picture) in python.

scaled_point_x = e[0] * sf[0]
scaled_point_y = e[1] * sf[0]
ee=[scaled_point_x,scaled_point_y]  # yellow color line in the figure

After adding the coordinate of center point to translate to the red point (center point), I get black line, which is not correct

new=[scaled_point_x+center_point[0],scaled_point_y+center_point[1]]

How can I fix this? which part am I doing wrong?

Answer 1

This is rather a math question than a programming question. To scale a point e by a factor f with respect to a center point cp ,

new_e = f*(e-cp)+cp

ie you scale the difference vector between the point and the center point and then translate it back to the center.

Answer 2

This sort of problem maybe tackled in chapter 2 of computer graphics books.

Try this,

1. translate the center_point to origin ie subtract centre_point from point
2. scale the point ie multiply by sf
3. translate the center_point to original position ie add center_point to point

here is some python

scaled_pts = []
for s in sf:
    tr_pointx, tr_pointy = e[0]-center_point[0], e[1]-center_point[1]
    sc_pointx, sc_pointy = tr_pointx * s, tr_pointy * s
    scaled_pt = [sc_pointx + center_point[0], sc_pointy + center_point[1]]
    # draw the pt  
    scaled_pts.append(scaled_pt)