In your second case this pattern doesn't have to express the future. For example, "He was the first man to climb Everest". There are examples like "Here is a computer to suit your needs" or "It was a dress to die for" which don't seem to fit any of your categories.
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park sang joon I punched Hulk in the face that mocked me.That doesn't seem a very natural sentence to me.
park sang joonI think adjective to-infinitive phrases can't replace with "that" clauses unless "that" clauses modify a abstract noun in certain category, a noun modified by the superlative degree or ordinal number and express the futureYou need to re-think this. Here is a good summary of the various uses of the to-infinitive:
park sang joonI'd like to know abstract nouns only to-infinitive phrases can modify, not relative "that" clauses or "of~ing" phrases.I doubt there is any noun that cannot be modified by a "that" clause. If you can think of one, please share it.
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park sang joonThen, I'd like to know whether I can't rephrase AlpheccaStars's example to the following.1. You have the opportunity to go on the trip of your life!2. He had an intense desire to succeed.3. Sarah felt a desperate need to run.1-1. You have the opportunity that I can go on the trip of your life!2-1. He had an intense desire that he would succeed.3-1. Sarah fel