Brassia hybrids are what immediately spring to mind when I think of pointy orchids whose flowers still have the 'typical orchid shape'
You can find many varieties of color and pattern in oncidium intergenic hybrids that have brassia in their parentage. Like
this gallery.
Another idea would be the angraecum family. These tend to be white waxy flowers with a starlike shape.
This blog has nice photography showcasing many different angraecum species.
Check out
angraecum sesquipedale, which is also known as darwin's orchid. It's a cool looking orchid that also has a nice little story behind the speciess' discovery.
You might also be interested in the
ghost orchid
The above all have more or less the typical orchid shape. If you're not set on that, then there's a whole world of sharp-petaled orchids out there.
For example,
habinara medusae. Many masdevallias and bulbophylum orchids will also have elongated petals that taper to a sharp point.
Finally, you can browse the IOSPE
orchid photos database to find many orchid types that are rarely if ever cultivated, and thus rarely discussed on sites like this one. There's many thousands of species to browse through, but many extremely odd and unique looking flowers to be found, if you have the time and patience for it.
Edited for clarity