imaginary curve

{{Short description|Algebraic curve}}

In algebraic geometry an imaginary curve is an algebraic curve which does not contain any real points.{{citation

| last = Petrowsky | first = I.

| doi = 10.2307/1968723

| issue = 1

| journal = Annals of Mathematics

| mr = 1503398

| pages = 189–209

| series = Second Series

| title = On the topology of real plane algebraic curves

| volume = 39

| year = 1938| jstor = 1968723

}}.

For example, the set of pairs of complex numbers $(x,y)$ satisfying the equation $x^2+y^2=-1$ forms an imaginary circle, containing points such as $(i,0)$ and $(\backslash frac\{5i\}\{3\},\backslash frac\{4\}\{3\})$ but not containing any points both of whose coordinates are real.

In some cases, more generally, an algebraic curve with only finitely many real points is considered to be an imaginary curve. For instance, an imaginary line is a line (in a complex projective space) that contains only one real point.{{citation

| last = Patterson | first = B. C.

| doi = 10.2307/2303867

| journal = The American Mathematical Monthly

| mr = 0006034

| pages = 589–599

| title = The inversive plane

| volume = 48

| year = 1941| issue = 9

| jstor = 2303867

}}.