Paths on a surface 
Game on the plane with a point removed
Instructions
After choosing two paths, the goal is to continuously deform one path into the other without crossing the point \(P\) (in red - the point removed from the plane).
On the vertical bar, below the figure, we can choose the desired action to proceed:
- [reset]: Eliminates any choice made over the paths.
- Choosing paths: We can move freely the marked points on the paths, to determine their configurations, but we cannot go over the point \(P\) (as this point is removed from the plane). If we disrespect this rule, we have to return to the option above [reset]. In alternative, we can choose, on the vertical bar at the right, one of the suggested paths.
- Start: Initial configuration of the game. If we choose the paths independently, a segment is added at the end up to the last common point.
- Play: Over one of the paths there are white marked points that we can manipulate. Drag the points to deform one path into the other. Once again, we cannot pass any arc of the path over the point \(P\); if we disrespect this rule, we will have to restart the game.
- Answer: At the app's bottom right corner, there is the indication if the chosen paths are homotopic or not. In case we choose one of the suggested examples, we can see one deformation of one path into the other (whenever possible); a vertical bar appears on the right of the paths, where we can move a point which controls the animation.
Remark on the game:
Removing a point of the plane makes certain moves impossible. Is the deformation always possible, independently of the paths and of the removed point? After the paths are chosen, we can see below each of them the number of number of times around \(P\) go the closed paths with respect to the paths completed by an arc from the start to the end point of each. From comparing the numbers we can determine if there is a solution for the game, with respect to each choice of paths. Let us explore how!
Removing a point of the plane makes certain moves impossible. Is the deformation always possible, independently of the paths and of the removed point? After the paths are chosen, we can see below each of them the number of number of times around \(P\) go the closed paths with respect to the paths completed by an arc from the start to the end point of each. From comparing the numbers we can determine if there is a solution for the game, with respect to each choice of paths. Let us explore how!
