Path homotopy
Concept history
A revision trail for this concept page.
Revision 1506
8/17/2026, 3:39:50 PM · Ancient Tree
Shared settings updated from fr translation
browser visibilityHidden from the concept browserListed in the concept browser
Revision 1257
8/13/2026, 10:43:02 AM · Ancient Tree
Updated linked exercises
linked exercisesNoneA simple path homotopy
Revision 1250
8/13/2026, 9:55:55 AM · Ancient Tree
Concept saved without content changes
No content or metadata changes were recorded.
Revision 1249
8/13/2026, 9:52:21 AM · Ancient Tree
Updated text
Compare with revision 12484 changed lines
1
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]2
$$H:[0,1]\times[0,1]\rightarrow X$$3
such that the endpoints stay fixed:4
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$5
6
```jsxgraph7
{8
"boundingBox": [-3.2, 2.2, 3.2, -2.0],9
"axis": false,10
"height": 400,11
"elements": [12
{13
"id": "s",13
"id": "t",14
"type": "slider",15
"parents": [[-2.8, -1.7], [0.2, -1.7], [0, 0.5, 1]],16
"attributes": {17
"name": "t"18
}19
},20
21
{22
"type": "functiongraph",23
"parents": ["0.425*x + 0.05 + 0.9*(1 - x^2/4)", -2, 2],24
"attributes": {25
"strokeColor": "#2f6f4e",26
"strokeWidth": 327
}28
},29
30
{31
"type": "functiongraph",32
"parents": ["0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4)", -2, 2],33
"attributes": {34
"strokeColor": "#e34a4a",35
"strokeWidth": 336
}37
},38
39
{40
"type": "functiongraph",41
"parents": [42
"(1-t.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + a.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",42
"(1-t.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + t.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",43
-2,44
245
],46
"attributes": {47
"strokeColor": "#3366cc",48
"strokeWidth": 449
}50
},51
52
{53
"type": "point",54
"parents": [-2, -0.8],55
"attributes": {56
"size": 3,57
"fixed": true,58
"withLabel": false59
}60
},61
62
{63
"type": "point",64
"parents": [2, 0.9],65
"attributes": {66
"size": 3,67
"fixed": true,68
"withLabel": false69
}70
}71
]72
}73
```74
75
##### Remarks76
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.Revision 1248
8/13/2026, 9:51:39 AM · Ancient Tree
Updated text
Compare with revision 12472 changed lines
1
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]2
$$H:[0,1]\times[0,1]\rightarrow X$$3
such that the endpoints stay fixed:4
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$5
6
```jsxgraph7
{8
"boundingBox": [-3.2, 2.2, 3.2, -2.0],9
"axis": false,10
"height": 400,11
"elements": [12
{13
"id": "s",14
"type": "slider",15
"parents": [[-2.8, -1.7], [0.2, -1.7], [0, 0.5, 1]],16
"attributes": {17
"name": "t"18
}19
},20
21
{22
"type": "functiongraph",23
"parents": ["0.425*x + 0.05 + 0.9*(1 - x^2/4)", -2, 2],24
"attributes": {25
"strokeColor": "#2f6f4e",26
"strokeWidth": 327
}28
},29
30
{31
"type": "functiongraph",32
"parents": ["0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4)", -2, 2],33
"attributes": {34
"strokeColor": "#e34a4a",35
"strokeWidth": 336
}37
},38
39
{40
"type": "functiongraph",41
"parents": [42
"(1-a.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + a.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",42
"(1-t.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + a.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",43
-2,44
245
],46
"attributes": {47
"strokeColor": "#3366cc",48
"strokeWidth": 449
}50
},51
52
{53
"type": "point",54
"parents": [-2, -0.8],55
"attributes": {56
"size": 3,57
"fixed": true,58
"withLabel": false59
}60
},61
62
{63
"type": "point",64
"parents": [2, 0.9],65
"attributes": {66
"size": 3,67
"fixed": true,68
"withLabel": false69
}70
}71
]72
}73
```74
75
##### Remarks76
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.Revision 1247
8/13/2026, 9:51:17 AM · Ancient Tree
Updated text
Compare with revision 12462 changed lines
1
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]2
$$H:[0,1]\times[0,1]\rightarrow X$$3
such that the endpoints stay fixed:4
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$5
6
```jsxgraph7
{8
"boundingBox": [-3.2, 2.2, 3.2, -2.0],9
"axis": false,10
"height": 400,11
"elements": [12
{13
"id": "s",14
"type": "slider",15
"parents": [[-2.8, -1.7], [0.2, -1.7], [0, 0.5, 1]],16
"attributes": {17
"name": "a"17
"name": "t"18
}19
},20
21
{22
"type": "functiongraph",23
"parents": ["0.425*x + 0.05 + 0.9*(1 - x^2/4)", -2, 2],24
"attributes": {25
"strokeColor": "#2f6f4e",26
"strokeWidth": 327
}28
},29
30
{31
"type": "functiongraph",32
"parents": ["0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4)", -2, 2],33
"attributes": {34
"strokeColor": "#e34a4a",35
"strokeWidth": 336
}37
},38
39
{40
"type": "functiongraph",41
"parents": [42
"(1-a.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + a.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",43
-2,44
245
],46
"attributes": {47
"strokeColor": "#3366cc",48
"strokeWidth": 449
}50
},51
52
{53
"type": "point",54
"parents": [-2, -0.8],55
"attributes": {56
"size": 3,57
"fixed": true,58
"withLabel": false59
}60
},61
62
{63
"type": "point",64
"parents": [2, 0.9],65
"attributes": {66
"size": 3,67
"fixed": true,68
"withLabel": false69
}70
}71
]72
}73
```74
75
##### Remarks76
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.Revision 1246
8/13/2026, 9:50:05 AM · Ancient Tree
Updated text
Compare with revision 124543 changed lines
1
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]2
$$H:[0,1]\times[0,1]\rightarrow X$$3
such that the endpoints stay fixed:4
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$5
6
```jsxgraph7
{8
"boundingBox": [-3.5, 2.5, 3.5, -2.5],8
"boundingBox": [-3.2, 2.2, 3.2, -2.0],9
"axis": false,10
"height": 400,11
"elements": [12
{13
"type": "line",14
"parents": [[-4, 0], [4, 0]],15
"attributes": {16
"strokeColor": "#aaaaaa",17
"strokeWidth": 118
}19
},20
{21
"type": "line",22
"parents": [[0, -3], [0, 3]],23
"attributes": {24
"strokeColor": "#aaaaaa",25
"strokeWidth": 126
}27
},28
29
{30
"id": "s",31
"type": "slider",32
"parents": [[-3, -2], [0, -2], [0, 0.5, 1]],15
"parents": [[-2.8, -1.7], [0.2, -1.7], [0, 0.5, 1]],33
"attributes": {34
"name": "a"35
}36
},37
38
{39
"type": "functiongraph",40
"parents": ["1-x^2/4", -2, 2],23
"parents": ["0.425*x + 0.05 + 0.9*(1 - x^2/4)", -2, 2],41
"attributes": {42
"strokeColor": "#2f6f4e",43
"strokeWidth": 344
}45
},46
47
{48
"type": "functiongraph",49
"parents": ["-1+x^2/4", -2, 2],32
"parents": ["0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4)", -2, 2],50
"attributes": {51
"strokeColor": "#e34a4a",52
"strokeWidth": 353
}54
},55
56
{57
"type": "functiongraph",58
"parents": ["(1-2*a.Value())*(1-x^2/4)", -2, 2],41
"parents": [42
"(1-a.Value())*(0.425*x + 0.05 + 0.9*(1 - x^2/4)) + a.Value()*(0.425*x + 0.05 - 0.75*(1 - x^2/4) + 0.35*x*(1 - x^2/4))",43
-2,44
245
],59
"attributes": {60
"strokeColor": "#3366cc",61
"strokeWidth": 462
}63
},64
65
{66
"type": "point",67
"parents": [-2, 0],54
"parents": [-2, -0.8],68
"attributes": {69
"name": "A",70
"size": 3,71
"fixed": true57
"fixed": true,58
"withLabel": false72
}73
},74
75
{76
"type": "point",77
"parents": [2, 0],64
"parents": [2, 0.9],78
"attributes": {79
"name": "B",80
"size": 3,81
"fixed": true67
"fixed": true,68
"withLabel": false82
}83
}84
]85
}86
```87
88
##### Remarks89
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.Revision 1245
8/13/2026, 9:48:04 AM · Ancient Tree
Updated text
Compare with revision 124482 changed lines
1
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]2
$$H:[0,1]\times[0,1]\rightarrow X$$3
such that the endpoints stay fixed:4
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$5
6
```jsxgraph7
{8
"boundingBox": [-3.5, 2.5, 3.5, -2.5],9
"axis": false,10
"height": 400,11
"elements": [12
{13
"type": "line",14
"parents": [[-4, 0], [4, 0]],15
"attributes": {16
"strokeColor": "#aaaaaa",17
"strokeWidth": 118
}19
},20
{21
"type": "line",22
"parents": [[0, -3], [0, 3]],23
"attributes": {24
"strokeColor": "#aaaaaa",25
"strokeWidth": 126
}27
},28
29
{30
"id": "s",31
"type": "slider",32
"parents": [[-3, -2], [0, -2], [0, 0.5, 1]],33
"attributes": {34
"name": "a"35
}36
},37
38
{39
"type": "functiongraph",40
"parents": ["1-x^2/4", -2, 2],41
"attributes": {42
"strokeColor": "#2f6f4e",43
"strokeWidth": 344
}45
},46
47
{48
"type": "functiongraph",49
"parents": ["-1+x^2/4", -2, 2],50
"attributes": {51
"strokeColor": "#e34a4a",52
"strokeWidth": 353
}54
},55
56
{57
"type": "functiongraph",58
"parents": ["(1-2*a.Value())*(1-x^2/4)", -2, 2],59
"attributes": {60
"strokeColor": "#3366cc",61
"strokeWidth": 462
}63
},64
65
{66
"type": "point",67
"parents": [-2, 0],68
"attributes": {69
"name": "A",70
"size": 3,71
"fixed": true72
}73
},74
75
{76
"type": "point",77
"parents": [2, 0],78
"attributes": {79
"name": "B",80
"size": 3,81
"fixed": true82
}83
}84
]85
}86
```87
6
##### Remarks7
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.Revision 1244
8/13/2026, 9:46:26 AM · Ancient Tree
Concept created
A path homotopy between two [[Path in a space|paths]] $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a [[Homotopy|homotopy]]
$$H:[0,1]\times[0,1]\rightarrow X$$
such that the endpoints stay fixed:
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$
##### Remarks
- If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.