Save by clicking here (with the right mouse button) the file on your computer (change the name if you want, but leave ".htm").
At this point you can use it to calculate the distance between any pair of curves.
Just open the source code, edit x1(t) and y1(t), x2(t) and y2(t), and run "save".
You can repeat this several times.

 

First example.

The distance between the curve x=cos(t)+2, y=sin(t)*1.5+2.5
and the curve x=cos(t)*1.5+4.5, y=sin(t)+5:

distance = 0.986024149
nearest point2 of the curves: (2.554700, 3.748075), (3.251925, 4.4452998)
    

function x1(t) { with(Math) {
return  cos(t)+2
}}
function y1(t) { with(Math) {
return  sin(t)*1.5+2.5
}}
function x2(t) { with(Math) {
return  cos(t)*1.5+4.5
}}
function y2(t) { with(Math) {
return  sin(t)*1+5
}}

 

We can draw the graph with JavaScript; for example, see the script curve_curve  (see the code;  se also here).

Moreover, the graph can be drawn, as well as with R, with Desmos:


 

Other curves

The distance between  y = x²  and  x = cos(t)+2, y = sin(t)*1.5-2.5:

distance = 1.74943908401
nearest point2 of the curves: (1.520061, -1.184047), (0.410297, 0.1683436)
   

function x1(t) { with(Math) {
return  t
}}
function y1(t) { with(Math) {
return  pow(t,2)
}}
function x2(t) { with(Math) {
return  cos(t)+2
}}
function y2(t) { with(Math) {
return  sin(t)*1.5-2.5
}}
d = 1.7494390840149902  t1 = 0.41029702324367284  t2 = 2.0713815877036205
Q1: 0.41029702324367284,0.16834364728261902  Q2: 1.520060928766358,-1.1840472661289747
  a1,b1: 0.41029,0.4103  a2,b2: 2.0713,2.0714
d = 1.7494390840150015  t1 = 0.4102970454827357  t2 = 2.0713816899039496
Q1: 0.4102970454827357,0.16834366553186208  Q2: 1.5200608391058256,-1.184047339703878
  a1,b1: 0.4102,0.4103  a2,b2: 2.071,2.072
d = 1.749439084691748  t1 = 0.41028668402882107  t2 = 2.0714091751288546
Q1: 0.41028668402882107,0.16833516309136565  Q2: 1.5200367264505558,-1.1840671270546023
  a1,b1: 0.405,0.415  a2,b2: 2,2.1
d = 1.7494390851059962  t1 = 0.4102815903498466  t2 = 2.0714163627364996
Q1: 0.4102815903498466,0.16833098337999935  Q2: 1.520030420856835,-1.1840723017701355
  a1,b1: 0.35,0.45  a2,b2: 1.8,2.2
d = 1.7494391666857405  t1 = 0.4105189163597722  t2 = 2.071121855292903
Q1: 0.4105189163597722,0.16852578068920165  Q2: 1.5202888086696688,-1.1838603269208176
  a1,b1: 0,5  a2,b2: 0,6.28

Other curves

The distance between  y = x/2;  and  x = cos(t)+2, y = sin(t)*1.5-2.5:

distance = 1.71628160613
nearest point2 of the curves: (0.9162278, 0.4581139), (1.6837722, -1.0769750)

   

function x1(t) { with(Math) {
return  t
}}
function y1(t) { with(Math) {
return  t/2
}}
function x2(t) { with(Math) {
return  cos(t)+2
}}
function y2(t) { with(Math) {
return  sin(t)*1.5-2.5
}}
d = 1.7162816061266104  t1 = 0.9162277661823271  t2 = 1.8925468740338862
Q1: 0.9162277661823271,0.4581138830911636  Q2: 1.6837722407735076,-1.0769750495290564
  a1,b1: 0.91622,0.91623  a2,b2: 1.89254,1.89255
d = 1.7162816061266106  t1 = 0.9162277629298269  t2 = 1.892546883010431
Q1: 0.9162277629298269,0.45811388146491344  Q2: 1.6837722322576094,-1.0769750537870057
  a1,b1: 0.9162,0.9163  a2,b2: 1.8925,1.8926
d = 1.7162816061304648  t1 = 0.9162238576869414  t2 = 1.8925479585265916
Q1: 0.9162238576869414,0.4581119288434707  Q2: 1.6837712119335746,-1.0769755639499405
  a1,b1: 0.91,0.92  a2,b2: 1.89,1.9
d = 1.7162816116011616  t1 = 0.9161148453240433  t2 = 1.892537416791737
Q1: 0.9161148453240433,0.45805742266202165  Q2: 1.6837812127153428,-1.076970563628954
  a1,b1: 0.8,1  a2,b2: 1.8,2
d = 1.7162829284602516  t1 = 0.9184100143789129  t2 = 1.8921826600372027
Q1: 0.9184100143789129,0.45920500718945645  Q2: 1.6841177854763587,-1.0768023820522356
  a1,b1: -1,3  a2,b2: 0,6.28

Other curves

The distance between  y² = 4·x;  and the circle whose center is (6,0) and radius is √5.
Or:  between  x = y²/4  and  x = cos(t)·√5+6, y = sin(t)·√5:

distance = √5
nearest points of the curves:  (4, 4), (5, 2), or:  (4, -4), (5, -2)

   

function x1(t) { with(Math) {
return  pow(t,2)/4
}}
function y1(t) { with(Math) {
return  t
}}
function x2(t) { with(Math) {
return  cos(t)*sqrt(5)+6
}}
function y2(t) { with(Math) {
return  sin(t)*sqrt(5)
}}
d = 2.2360679774997893  t1 = 4  t2 = 2.034443942014127
Q1: 4,4  Q2: 4.999999987563151,1.9999999937815757
  a1,b1: 4,4  a2,b2: 2.03444,2.03445
d = 2.2360679774997893  t1 = 4  t2 = 2.034443944439723
Q1: 4,4  Q2: 4.99999998271196,1.99999999135598
  a1,b1: 4,4  a2,b2: 2.034,2.035
d = 2.23606797749979  t1 = 4  t2 = 2.034443922597264
Q1: 4,4  Q2: 5.000000026396878,2.000000013198439
  a1,b1: 4,4  a2,b2: 2,2.1
d = 2.236067984081465  t1 = 3.9999248114594516  t2 = 2.0345241064510655
Q1: 3.9998496243322323,3.9999248114594516  Q2: 4.999839661903113,1.9999198229173891
  a1,b1: 3.9,4.1  a2,b2: 1.9,2.1
d = 2.23607357581484  t1 = 3.999332297031233  t2 = 2.0332239002674015
Q1: 3.998664705519279,3.999332297031233  Q2: 5.00244081469452,2.001218546739128
  a1,b1: 0,10  a2,b2: 0,6.28

Other curves

The distance between  y = x+10;  and the parabola  y² = 36·x  (x = y²/36):

distance = √2/2
nearest points of the curves:  (8.5, 18.5), (9, 18)


 
   

function x1(t) { with(Math) {
return  t
}}
function y1(t) { with(Math) {
return  t+10
}}
function x2(t) { with(Math) {
return  pow(t,2)/36
}}
function y2(t) { with(Math) {
return  t
}}
d = 0.7071067811865476  t1 = 8.5  t2 = 18
Q1: 8.5,18.5  Q2: 9,18
  a1,b1: 8.5,8.5  a2,b2: 18,18
d = 0.707106782444305  t1 = 8.50006113808483  t2 = 18.000031548471813
Q1: 8.50006113808483,18.50006113808483  Q2: 9.00003154849946,18.000031548471813
  a1,b1: 8.4,8.6  a2,b2: 17.9,18.1
d = 0.7071068277279571  t1 = 8.500758554813238  t2 = 18.000905266627672
Q1: 8.500758554813238,18.50075855481324  Q2: 9.000905289391774,18.000905266627672
  a1,b1: 8,9  a2,b2: 17,19
d = 0.7071068199893746  t1 = 8.498598716340616  t2 = 17.998640926050758
Q1: 8.498598716340616,18.498598716340616  Q2: 8.99864097735859,17.998640926050758
  a1,b1: 5,10  a2,b2: 10,20
d = 0.7071296055077205  t1 = 8.482876301283491  t2 = 17.986566077927918
Q1: 8.482876301283491,18.48287630128349  Q2: 8.986571090990757,17.986566077927918
  a1,b1: -5,20  a2,b2: 0,25