～ている間(あいだ)にWhile, During the time that ~

Both forms involve the idea of an ongoing process, but with different nuance and focus. ているあいだに refers to something happening while another action or state is ongoing. For example, 勉強しているあいだに電話がかかってきた ('The phone rang while I was studying.'). In contrast, つつある expresses something in the process of changing, e.g. 病気が治りつつある ('My illness is in the process of healing.').

間(あいだ) means 'period' or 'space', and in addition to its literal meaning, it is used in expressions meaning 'while' or 'between', e.g. 1月と4月の間に 'between January and April'. In the ているあいだに form, the focus is on something happening during another ongoing action. For example, 勉強しているあいだに電話がかかってきた ('The phone rang while I was studying.').

Both patterns express being in the middle of an action, but differ in how they are used. 中 attaches to nouns to indicate 'in the process of' or 'during.' For example, 今、食事中です means 'I'm in the middle of eating.' ているあいだに refers to something happening while another action or state is ongoing. For example, ご飯を食べているあいだに電話が鳴った and 食事中に電話が鳴った are equivalent in meaning.

とき expresses the time of an action or event, like the English 'when'. It can be used with nouns as in 雨の時 'when it rains', verbs as in 父が帰ってきた時 'when Dad came home', or adjectives as in 暑い時 'when it's hot'. In contrast, ているあいだに uses the て-form of verbs and refers to a period during which something occurs, e.g. 晩ご飯を食べている間に、父が帰ってきました ('While I was eating dinner, my father came home.').

Both patterns express ongoing actions but differ in focus. ているところだ means being in the middle of something, often with a sense of being occupied. For example, 勉強をしているところだ ('I'm in the middle of studying.').ているあいだに highlights something that happens during another action, e.g. 勉強しているあいだに電話がかかってきた ('The phone rang while I was studying.').

Both forms involve the idea of simultaneous or ongoing actions, but differ in structure and nuance. ているあいだに expresses that something happened during another ongoing action, e.g. 勉強しているあいだに電話がかかってきた ('The phone rang while I was studying.'). ながら shows one subject doing two things at once, e.g. 音楽を聴きながら勉強するのが好き ('I like to study while listening to music.').

Both patterns can express the idea of 'while,' but differ in nuance. ているあいだに refers to something happening during another ongoing action. For example, 勉強しているあいだに電話がかかってきた means 'The phone rang while I was studying.' うちに emphasizes doing something while a particular state still lasts. For example, 早いうちに帰ろう means 'Let's go home while it's still early.'

Both can mean 'while,' but the nuance differs. ついでに implies doing something extra during another task, e.g. 買い物のついでに郵便局に寄った ('While shopping, I stopped at the post office.'). ているあいだに just marks that something happened during another action, as in 買い物しているあいだに電話がかかってきた ('I got a call while shopping.').

The ている① form describes an ongoing action or state. When combined with あいだに (meaning 'while doing something'), it expresses the idea of 'while doing A, B happened.' This is typically used for events outside the speaker's control. For example, 映画を見ている間に、電話が掛かってきました means 'While I was watching a movie, a phone call came in.'

Both express something happening during an ongoing action, but differ in focus. Compared to ているあいだに, which states in a straightforward manner one thing happening in the middle of another thing, ～ところに・～ところへplaces more of a focus on the 'scene' or situation, and is therefore often used to express a surprising or disturbing event.