Why does my twin flame avoid meeting me in person? We have constant chat, telephone, and video contact.

My dear, be thankful that at least you communicate with each other. It is indeed a blessing to have some sort of contact with him or her.
Here are some of the reasons why a Twin Flame is afraid to meet you in person:
Insecurities —
A Twin Flame will avoid you at all costs because you are the trigger of all his or her frustration. Do not take this personally though. He or she has issues to deal with. You are not the source of their pain.

I advise you to be open and honest with your Twin Flame; call him or her out if they are hiding from their pain. It is best for both of you to be grounded in your authentic selves — it will save you so much frustration.

Your separation cycles will continue if you are unwilling to confront the nature of your Twin Flame dynamic.

Self-rediscovery is a process

When a Twin Flame is dealing with his or her issues, they need that space to find themselves at his or her pace.

A Twin Flame does not want to inflict his pain on you. It is best to deal with his or he issues in private.

You are also another excuse for them to delay the progress of their healing journey. A Twin Flame can hide behind you or blame you for their issues.
Respect their need for space.