Return to Answer

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get four real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

You can use this interactive web widget to experiment with this setup. It doesn't do all the degenerate cases yet, but the behavior in a small neighborhood around such a singular situation should give you a fair idea of what to expect in the limit.

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get four real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

You can use this interactive web widget to experiment with this setup. It doesn't do all the degenerate cases yet, but the behavior in a small neighborhood around such a singular situation should give you a fair idea of what to expect in the limit.

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

You can use this interactive web widget to experiment with this setup. It doesn't do all the degenerate cases yet, but the behavior in a small neighborhood around such a singular situation should give you a fair idea of what to expect in the limit.

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get four real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

You can use this interactive web widget to experiment with this setup. It doesn't do all the degenerate cases yet, but the behavior in a small neighborhood around such a singular situation should give you a fair idea of what to expect in the limit.

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

In general there will be four solutions, possibly complex. In Conic by three points and two tangent lines I've asked for ways to compute these, and based on a comment there found a way which is in my opinion pretty intuitive to understand since it relates to a 3d setup you can easily visualize. You can also use this to count possible solutions.

So when will the solutions be complex not real? Well, consider the two lines dividing the plane into two regions. In an affine setup that would be four regions, but they connect through infinity so there are only two. If all of the points are in the same region, you get real solutions. If they are in different regions, you only get complex solutions.

If your elements are not in general position, there are further setups to consider. If the three points are collinear and within one region, then you have only a single conic (since all the planes in the idea behind the computation will coincide). If one point coincides with one of the lines, you only get two solutions. If two points coincide with the two lines you only get one solution. And I don't make any claims to completeness here at this point.

You can use this interactive web widget to experiment with this setup. It doesn't do all the degenerate cases yet, but the behavior in a small neighborhood around such a singular situation should give you a fair idea of what to expect in the limit.