ところを

1. 1Politely acknowledges someone’s circumstances while thanking or apologizing to them.Ex. 1
2. 2Marks what someone was doing or about to do when another person acted on them.Ex. 2

The first use is polite and occurs in expressions of thanks or apology; the second can occur in neutral narration, often with a verb such as 呼び止める. With ところに / ところへ, the emphasis is on something happening or arriving at that moment; ところを presents the person’s situation as the setting in which they are affected by the following action.

• ところに / ところへN2With ところに / ところへ, the following clause is an external event that happens to occur, such as a visit or a phone call; with ところを, it is an action directed at that very scene (e.g., being seen or being called out to) or a polite expression of thanks or apology. They generally cannot be interchanged.
• 最中にN3最中に emphasizes that something happens by chance to interrupt an activity in progress, and it attaches only to 「名詞＋の」 or the 「ている」 form. The second clause of ところを must directly target the scene (being seen, being called out, etc.), and it also has polite uses; they generally cannot be interchanged.
• を押してN1を押して attaches only to nouns expressing obstacles, such as 「反対」 or 「病」, and means the actor pushes through the obstacle and goes ahead anyway; it is a written-language expression. ところを points to a particular scene and is used for polite expressions or for being seen, being called out, etc.; they cannot be interchanged.
• ところだったN2ところだった is mostly used at the end of a sentence and indicates that something almost happened (but in fact did not) or that one was just about to do something at the time. ところを must be followed by an action directed at the scene (being seen, being called out, etc.) or a polite word of thanks or apology; the two cannot be interchanged.
• のにN4のに is adversative — the result is contrary to what was expected, often with dissatisfaction or regret — and it can be used at the end of a sentence. ところを does not express contrast; it treats the scene at that moment as the object of the following action, used in polite expressions or for 'being seen, being called out,' etc., so the two cannot be interchanged.

The second clause of ところを is an action that directly acts on the scene (seeing, catching, calling out to someone), while ところに／ところへ indicates that someone arrives or something happens at that moment. 最中に indicates that another event occurs while something is in progress; を押して means forcing oneself to act despite illness or opposition; ところだった means something almost happened; and のに is adversative, often with a nuance of surprise or regret, but does not indicate 'right at that scene.'