how to find the equation of one curve in XY plan which satisfies such functions?

Question

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows:

1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the curve we want to find;

2) $f(x,y)$ pass A and B;

3) the tangent vector of $f$ at A is $v_A=(v_{ax},v_{ay})$ and at B is $v_B=(v_{bx},v_{by})$, and $||v_A||=||v_B||=1$.

I think that there is a group of curves which satisfy these conditions, so I want to add one more condition:

4) the length of $f $ from A to B or from B to A is the minimum.

Now how many curves are there? and how to get the equations? If you want, you can answer with a numercial example. For example, $A=(0,0),B=(1,2),f'(A)=(-1,0),f'(B)=(0,1)$.

This problem stops my work a long time, so thanks for your help.

Answer 1

First you have 2 given points and the direction the car is facing:

Then draw the 2 tangent circles to each position with a radius given by the turn radius of the car:

Then find the line segments that are tangent two circles, one from each given point. Only 3 are shown here, but since each circle contributes two locations, there are a total of 16 tangent line segments:

Then find the length of each of the 16 paths. A path is the car turning along the circle, following and tangent line segment to the other circle, and the finishing along the other circle. This is where you have to decide if the car is allowed to drive backwards. If not, some of the paths will be invalid as the car would have to travel backwards at some point. Blue is an example of one path:

The shortest of the paths is the shortest route the car can take. Note that the instant change from turning to straight isn't actually physically possible unless you stop the car and then turn the wheels and then start again, but there are a substantial number of details like that I mentioned earlier (friction, acceleration, top speed) that can make this problem hideous. However, it sounds like this is simple enough for your purposes.

Answer 2

Perhaps your question is similar to this one: Fastest curve from $p_0$ to $p_1$ ?

Answer 3

Just an idea:

It might be more suitable, to formulate the problem for a parametrized curve. This gives a variational problem for two functions $$\vec{r}(t)=(x(t),y(t))$$

with 8 side conditions $$\vec{r}(0)=(x_a,y_a), \qquad \vec{r}(1)=(x_b,y_b),$$ $$\dot{\vec{r}}(0)=(v_{ya},-v_{xa}),\qquad \dot{\vec{r}}(1)=(v_{yb},-v_{xb})$$ that minimizes the length integral

$$\int_0^1 dt \frac{dL}{dt}(t)$$

Answer 4

Unless the directions specified by $f'(A), f'(B)$ are in line with the straight line connecting the points I suspect that you can't find such a function. In particular, there seems to be a family of curves connecting the points, and satisfying your conditions but where the length of the family approaches that of the straight line. Of course, this means that the limiting function can't itself work.

Consider the following family: